Going to a different University for part of my PhD
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.