How important is institution prestige at the postdoc level? I once heard a piece of advice from an established researcher in my field: even if you don't want to stay there permanently, choose your first tenure-track institution wisely, since you won't be able to move to a more prestigious university -- only comparable or weaker ones.
I was wondering, how relevant is this at the postdoc level? Of course it's always better to have worked at high-tier universities, but e.g. would getting a postdoc job at a lower-ranked university (because I like the location, because of a good advisor, whatever) prevent me from getting another postdoc position at a higher-level institution? Would it decrease my chances of getting a TT position at a good place later on, even if I have previously worked at a prestigious institution?
(Just to clarify, I'm a non-US postdoc and planning to do at least a few postdoc stints before looking for more permanent positions. This might not apply to the US where I often see people only do one postdoc and go into tenure-track after that.)
&quot;choose your first tenure-track institution wisely, since you won't be
able to move to a more prestigious university -- only comparable or
weaker ones.&quot;

While I agree to choose your first tenure-track position wisely, I disagree with the reason that &quot;you won't be able to move to a more prestigious university&quot;. If you perform extremely well, more prestigious universities may find you quite desirable. I can give you a list of examples of people that moved &quot;up&quot; if you're interested.

&quot;would getting a postdoc job at a lower-ranked university (because I
like the location, because of a good advisor, whatever) prevent me
from getting another postdoc position at a higher-level institution?&quot;

It will not alone (with all else being equal) prevent you from getting a postdoc position at a higher institution. At the same time, moving to a place only because you like the location doesn't seem the best choice (in my opinion, though other opinions are equally valid). If you like the adviser, research project, and other things at a Tier 2 university so much that you would turn down an offer from a Tier 1 university because of it, then it sounds like the Tier 2 university is the right place for you. Just remember that the position at the Tier 1 university will often make it easier for you to move to other Tier 1 universities thanks to the connections you'll make there and the fact that people seeing your CV will know that already you passed the selection process by the first Tier 1 university (this sometimes carries more weight than one might imagine it should). It is certainly possible to do a first postdoc at a Tier 2 university, then a second postdoc or assistant professorship at a Tier 1 university (if you perform well!). It's also possible to go to a Tier 1 university and be the victim of an abusive supervisor that causes you life-long trauma and hate for academia, or you could also fail to survive the hyper-competitive climate of your peers.

&quot;Just to clarify, I'm a non-US postdoc and planning to do at least a
few postdoc stints before looking for more permanent positions.&quot;

If you're planning to do &quot;at least a few post-doc stints&quot;, then the first one being at a less prestigious place and the rest of them being at top places like Oxford and Cambridge does not sounds like a problem at all. Be careful that after too many post-doc positions you will become much less desirable for tenure-track positions even in non-US countries.
Even if you were to go to a Tier 4 or Tier 5 university, it can in some circumstances be okay: I can for example give you an example of a co-author who did his PhD at UCL (fairly prestigious in England), went to a third-world country to do his first post-doc at a place that I'd never heard of, returned to Europe to do a second post-doc at a respected university in Denmark, then got a faculty position in UK. His work was consistently of high quality, and was published in good journals, so it didn't matter much.
Finally, you ask about the difficulty of being hired for a postdoc position at a more prestigious university after doing a first postdoc at a less prestigious university, and I want to say: In my experience, getting into a prestigious university for a post-doc is very different from getting into one as a student. There's no exams. It's based on your publications, your work, your connections, and perhaps the impression you've made on people at conferences.