user10869: I'm wondering if filming with a 4K camera, possibly downsizing to Full HD, will give better detail than filming with a Full HD camera (other parameters being equals).
Sub-question (maybe I should make it a different question?):

I have some more experience and better equipement for photography (Canon 6D),
I want my video camera to behave well in low light in moving environments,
I'm often filming moving things/people, I don't want much bokeh/focus loss (I never really remember if that means more or less depth or field).

I currently own a Sony Cybershot RX-100 M2. Very unimpressive body size, but makes excellent 1920x1080x50p videos. I am thinking about an upgrade anyway, max budget for an outstanding camera would be 2000 EUR, but I am not sure if I want to put money in it right now since 4K market is still evolving fast, and so far I cannot really decide because other devices I got info about seem to be very similar or trade-off one advantage for another.
For example compared to Sony RX-100, Sony FDR-AX33 has a smaller sensor. Sony FDR-AX100 only opens at f/2.8 and I use for a wider aperture, so has far as I know RX-100 would have a better low light score than both those cameras. GH4 has more bokeh and seems kind of grainy in low light, ... and yet no camcorder in my budget has more than 30p with 4K.
Suggetions would be welcome, because I cannot really tell what would be a serious upgrade or just wasted money.

user10931: Short answer: no, 4K down-scaled to 1080p is not necessarily going to give you a better result than filming with a FullHD camera.
The SONY A7S is the hands-down best low-light video camera for less than $34,000.  (The Panasonic Varicam S35 and the Alexa can shoot better images, albeit at much higher costs).  Here's a surf video shot entirely by moonlight, which is just nuts: https://vimeo.com/108613669
But beware: often people say they want lowlight capabilities because they really just don't want to be bothered with lighting.  Most times, that means boring, flat-looking video.  Good lighting is what gives images depth and makes scenes pop.  If you blow off lighting, you cannot blame the camera for mediocre results.  Even dark scenes in movies are often very well lit: https://www.royalholloway.ac.uk/mediaarts/documents/pdf/lightingfordarkness.pdf
Not all FullHD cameras are created equal.  Canon's Cinema line (C100, C300, C500) have none of the pixel-skipping nonsense of the EOS5D3.  Their FullHD images are going to look much, much better than those that come from Canon DSLRs.  And of course, there are 4K (and up) cameras that are so excellent that yes, they will beat a FullHD camera all day long, because they are just plain better (and usually much more expensive).
For your pricepoint (2000 Euros), a good 4K camera will give you some nice footage that allows you to pull a decent still from the stream and post it as an image.  A good FullHD camera will likely give you better sensitivity, and thus theoretical better low-light performance.  But chances are those benefits are in the noise compared to all the other things you cannot do for 2000 EUR, such as proper lighting.

user10866: Yes, if you record a video in 4K, then downsize to 1080p, the image will look much clearer than if you shot the video on a 1080p camera. You can also zoom in up to 200% and still retain the quality in a 1080p time line.
Why it may look better is because most sensors these days use chroma subsampling. On most consumer 1080p video camera products, the camera records in 4:2:0. Meaning that each pixel actually shares it's color value with 3 similar surrounding pixels, but every pixel does get its own luminosity value (Y'). This looses 50% of the  information captured by 4:4:4, in which each pixel gets its own color value and luminosity value, but the size of the clips produced are 50% more. This is more expensive, and data-hogging.
But now in 4k video, every 1080p pixel now gets 4 individual pixels (it's still being recorded in 4:2:0 though), in which color values are still being shared however, when you downscale to 1080p, every pixel now gets its own individual color and luminosity value! This is called 4:4:4, which not to long ago, costed upwards of $10,000.
This video can explain this visually: Youtube: Why Does 4K Look Better on 1080p Monitors 
As to your sub-question about low-light performance, I am not sure if an identical spec camera (besides pixel count, of course) would give you better low-light performance or not, because that usually depends on the sensor size, aperture of the lens (assuming we're talking about DSLRs), and max ISO. If you just want a good low-light camera, check out the Sony A7S, as GiantCowFilms pointed out in his answer.

user6249: There are three key factors in your cameras low light performace:
1. Lens (Aperture)
The lower your f-stop, the more light gets through to the sensor, you will also have a very shallow (narrow) Depth of Field. Often this is a requirement at night, since cutting off most of your light is generally a bad idea. When looking at any camera, see what lenses are available to suit your needs.
2. "Pixel" Size
Each little diode on your sensor collects light, the size of each diode directly effects low-light performance. There are two significant factors in the size of diode: sensor size and resolution. If you have 1920 pixels across your sensor and your sensor is an inch wide each diode will be about 1/1920 of an inch across if you sensor is 3840 pixels across your diode will be about 1/3840 of an inch across, much much smaller, and therefor able to collect less light.
3. Sensitivity
The sensitivity of a sensor can be controlled with ISO. The max ISO is very important for low light capture. Some excellent cameras in other regards have very low max ISO's, and are therefor unable to shoot in low light. Some cameras have very high max ISO's, and are designed for low light. Such cameras are the Canon C-Hundreds series, and the spectacular Sony A7s. The A7s has a relatively low resolution sensor and very high resolution.
The simple answer is yes, 4K will negatively effect low light performance, but most cameras already have a 4k-sized sensor that they simply fail to utilize (the canon 6D suffers equally from small diode size as the A7s for example). In your case, no 4K will have no effect, but if you're going to get a 4K camera for low light, buy the A7s and an external recorder. Out of it you will get 4K raw. For about $3k. It will put your 6D's performance to shame. Some people have lit A7s shots with moonlight/an iphone. Gulp.. did you see that last one, it out did the C300..... :O!
PS:
I first mentioned lenses for a reason. The sony A7s takes Sony E mount lenses. In my opinion they are some of the worst selection of lenses on the market. (Some are good glass, but there are no versatile zooms). Be wary of the lenses, they are half the key to getting your night shot. Also you HD will look amazing with a good lens, they've shot Hollywood features in HD (like Scorsese's Hugo), utilizing the power of $30k lenses.

10869: <p>I'm wondering if <strong>filming with a 4K camera, possibly downsizing to Full HD, will give better detail than filming with a Full HD camera</strong> (other parameters being equals).</p>

<p>Sub-question (maybe I should make it a different question?):</p>

<ul>
<li>I have some more experience and better equipement for photography (Canon 6D),</li>
<li>I want my video camera to behave well in low light in moving environments,</li>
<li>I'm often filming moving things/people, I don't want much bokeh/focus loss (I never really remember if that means more or less depth or field).</li>
</ul>

<p><strong>I currently own a Sony Cybershot RX-100 M2.</strong> Very unimpressive body size, but makes excellent 1920x1080x50p videos. I am thinking about an upgrade anyway, max budget for an outstanding camera would be 2000 EUR, but I am not sure if I want to put money in it right now since 4K market is still evolving fast, and so far I cannot really decide because other devices I got info about seem to be very similar or trade-off one advantage for another.</p>

<p>For example compared to <strong>Sony RX-100</strong>, <strong>Sony FDR-AX33</strong> has a smaller sensor. Sony <strong>FDR-AX100</strong> only opens at f/2.8 and I use for a wider aperture, so has far as I know RX-100 would have a better low light score than both those cameras. <strong>GH4</strong> has more bokeh and seems kind of grainy in low light, ... and yet no camcorder in my budget has more than 30p with 4K.</p>

<p>Suggetions would be welcome, because I cannot really tell what would be a serious upgrade or just wasted money.</p>
 10931: <p>Short answer: no, 4K down-scaled to 1080p is not necessarily going to give you a better result than filming with a FullHD camera.</p>

<p>The SONY A7S is the hands-down best low-light video camera for less than $34,000.  (The Panasonic Varicam S35 and the Alexa can shoot better images, albeit at much higher costs).  Here's a surf video shot entirely by moonlight, which is just nuts: <a href="https://vimeo.com/108613669" rel="nofollow">https://vimeo.com/108613669</a></p>

<p>But beware: often people say they want lowlight capabilities because they really just don't want to be bothered with lighting.  Most times, that means boring, flat-looking video.  Good lighting is what gives images depth and makes scenes pop.  If you blow off lighting, you cannot blame the camera for mediocre results.  Even dark scenes in movies are often very well lit: <a href="https://www.royalholloway.ac.uk/mediaarts/documents/pdf/lightingfordarkness.pdf" rel="nofollow">https://www.royalholloway.ac.uk/mediaarts/documents/pdf/lightingfordarkness.pdf</a></p>

<p>Not all FullHD cameras are created equal.  Canon's Cinema line (C100, C300, C500) have none of the pixel-skipping nonsense of the EOS5D3.  Their FullHD images are going to look much, much better than those that come from Canon DSLRs.  And of course, there are 4K (and up) cameras that are so excellent that yes, they will beat a FullHD camera all day long, because they are just plain better (and usually much more expensive).</p>

<p>For your pricepoint (2000 Euros), a good 4K camera will give you some nice footage that allows you to pull a decent still from the stream and post it as an image.  A good FullHD camera will likely give you better sensitivity, and thus theoretical better low-light performance.  But chances are those benefits are in the noise compared to all the other things you cannot do for 2000 EUR, such as proper lighting.</p>
 10866: <p>Yes, if you record a video in 4K, then downsize to 1080p, the image will look much clearer than if you shot the video on a 1080p camera. You can also zoom in up to 200% and still retain the quality in a 1080p time line.</p>

<p>Why it may look better is because most sensors these days use chroma subsampling. On most consumer 1080p video camera products, the camera records in 4:2:0. Meaning that each pixel actually shares it's color value with 3 similar surrounding pixels, but every pixel does get its own luminosity value (Y'). This looses 50% of the  information captured by 4:4:4, in which each pixel gets its own color value and luminosity value, but the size of the clips produced are 50% more. This is more expensive, and data-hogging.</p>

<p>But now in 4k video, every 1080p pixel now gets 4 individual pixels (it's still being recorded in 4:2:0 though), in which color values are still being shared however, when you downscale to 1080p, every pixel now gets its own individual color and luminosity value! This is called 4:4:4, which not to long ago, costed upwards of $10,000.</p>

<p>This video can explain this visually: <a href="https://www.youtube.com/watch?v=kIf9h2Gkm_U" rel="nofollow">Youtube: Why Does 4K Look Better on 1080p Monitors </a></p>

<p>As to your <strong>sub-question</strong> about low-light performance, I am not sure if an identical spec camera (besides pixel count, of course) would give you better low-light performance or not, because that usually depends on the sensor size, aperture of the lens (assuming we're talking about DSLRs), and max ISO. If you just want a good low-light camera, check out the Sony A7S, as GiantCowFilms pointed out in his answer.</p>
 6249: <p>There are three key factors in your cameras low light performace:</p>
<h1>1. Lens (Aperture)</h1>
<p>The <em>lower</em> your f-stop, the more light gets through to the sensor, you will also have a very shallow (narrow) Depth of Field. Often this is a requirement at night, since cutting off most of your light is generally a bad idea. When looking at any camera, see what lenses are available to suit your needs.</p>
<h1>2. &quot;Pixel&quot; Size</h1>
<p>Each little diode on your sensor collects light, the size of each diode directly effects low-light performance. There are two significant factors in the size of diode: sensor size and resolution. If you have 1920 pixels across your sensor and your sensor is an inch wide each diode will be about 1/1920 of an inch across if you sensor is 3840 pixels across your diode will be about 1/3840 of an inch across, much much smaller, and therefor able to collect less light.</p>
<h1>3. Sensitivity</h1>
<p>The sensitivity of a sensor can be controlled with ISO. The max ISO is very important for low light capture. Some <a href="https://www.blackmagicdesign.com/products/cinemacameras" rel="nofollow noreferrer">excellent cameras</a> in other regards have very low max ISO's, and are therefor unable to shoot in low light. Some cameras have very high max ISO's, and are designed for low light. Such cameras are the Canon C-Hundreds series, and the spectacular Sony A7s. The A7s has a relatively low resolution sensor and very high resolution.</p>
<p>The simple answer is yes, 4K will negatively effect low light performance, <em>but</em> most cameras already have a 4k-sized sensor that they simply fail to utilize (the canon 6D suffers equally from small diode size as the A7s for example). In your case, no 4K will have no effect, but if you're going to get a 4K camera for low light, buy the A7s and an <a href="http://www.bhphotovideo.com/c/product/1046387-REG/atomos_atomshogun_shogun_7_4k_hdmi.html" rel="nofollow noreferrer">external recorder</a>. Out of it you will get 4K raw. For about $3k. It will put your 6D's performance to shame. Some people have lit A7s shots with <a href="https://vimeo.com/105690274" rel="nofollow noreferrer">moonlight</a>/<a href="https://youtu.be/gDprjGU9U64?t=8m14s" rel="nofollow noreferrer">an iphone</a>. Gulp.. did you see that last one, it out did the C300..... :O!</p>
<p><strong>PS:</strong></p>
<p>I first mentioned lenses for a reason. The sony A7s takes Sony E mount lenses. In my opinion they are some of the worst selection of lenses on the market. (Some are good glass, but there are no versatile zooms). Be wary of the lenses, they are half the key to getting your night shot. Also you HD will look amazing with a good lens, they've shot Hollywood features in HD (like Scorsese's Hugo), utilizing the power of $30k lenses.</p>
