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.