Going to a different University for part of my PhD SHORT: Is it normal -or perhaps "is there likely to be any kind of existing framework"- for me to go and do part of my PhD at another institution? For, say, a summer - although I might prefer something like 5 months or even a year?

DETAILS: I'll now try to explain why I'm thinking about this - while keeping details anonymous!

I am working on "X". I'm very happy to be doing so, but X is not quite a large enough thing to be the entire focus of a PhD - this is something I and my supervisor both believe. I am ~1.3 years into my 4 years of funding.

A few days ago I met "Dr Y" at a conference. With Dr Y, I conceived an idea/project "Z", that is very closely related to X. In my next supervisory meeting I will be talking to my supervisor about Z, and I believe it will excite him, because it fits so nicely into my PhD. So I am going to start working on Z soon; and if Z turns out really well, I might very well want to invest a year or maybe more out of my PhD into Z.

Thing is: Z is deeply integrated into Dr Y's work, but not that of my supervisor. My supervisor's work isn't even very related to X! Which I know is kinda bad, but we've been getting by ok.

However, if Z starts to turn into something cool, maybe you can see why I'd like to go off and work for Dr Y for a bit! He is in the same country but a different institution. Skype conversations and email can do the job but it's nice to be alongside the person whose work you are contributing to.

I believe Dr Y would be willing to make room for me if Z started to look promising. Not sure what my supervisor would think; but of course all of this is contingent on it all being an administrative possibility at all. So what's the verdict? 

Further information related to the comments

I feel like this is very different from "transfer for unhappy", because I'm not unhappy and I don't want to permanently transfer. I want to come back to work on project X. If nothing else, there's more paperwork with that, right?

"Ultimately your dissertation needs to be on something that matches the expertise of your supervisor, and that doesn't seem to be the case for Z": Well, it is not particularly the case for X either. 

I'd hope to continue using the funds given to me by my current funding source.

I'm based in the UK, but I can't be more specific.
As Tobias Kildetoft pointed out in a comment, in some places it is quite common -- and sometimes mandatory -- for a PhD student to spend a period in another group, possibly abroad, frequently paid by the home university (e.g., "study abroad" at my university). The visit period is usually from few months to one year, six months being probably quite common.

I personally consider such an experience an important part in the formation of a researcher.

There's usually no need for a framework. On the part of your home university, what counts are the regulation of your PhD programme, the opinion of your advisor and, sometimes, that of the PhD programme committee. The opinions of advisor and PhD programme committee usually depend on how much the activity in the other group is related to your current one or, more generally, to the topic of your PhD. On the part of the other university, the opinion of the group leader is usually the only thing that matters, especially if funding comes from the home university.

From your description, I think that you can certainly propose the visit to your advisor with chances for a positive outcome. You didn't provide enough details for us to tell you whether your university regulations allow this activity or not, but it's easy to check this with the university administration. Do this check before making the proposal.