Advisable to visit other labs?
I am actually currently just preparing for such a collaboration/internship/exchange (leaving after summer). From day 1 of my PhD (both) my supervisors said that we should definitely look for such an opportunity during my second year.
Both of my supervisor think it is not a bad idea, moreover, they both think it is something very important given several restrictions:

that you do it around the middle of your PhD.
This is important because in the beginning, you're still new to the topic and trying to get a grip on everything important. You can not &quot;give&quot; almost anything back to the lab you are visiting, so it is not a fair exchange.

the project (3 months in my case), is something directly related to my PhD
Not just &quot;has similarities&quot;, but maybe a sub-problem of the research direction I am currently on anyway due to my PhD. It is also important (mostly, for the grants), that it is something that the target laboratory is better suited for/equipped/has more experts than your own institution.

that it is around 3-4 months
The point is, much shorter than that, and it becomes a purely social / networking visit where you can possibly present your research in person and have other people present theirs, but not much more. On the other hand, more than 4 months will indeed be too long, as I am not expected to spend much more than that on a single idea even when I am working in my lab.


So, these were the restrictions. The benefits, however, are much more numerous:

obviously, networking. Especially if you plan to stay in academia, developing your own network of contacts as soon as possible is very important.
exchanging ideas. In addition to papers, presenting your own (and your labs) work in person is another way to disseminate ideas. Also, being exposed to different people, different approaches and different ideas might just jog some clogs that would otherwise take much longer to click in place.
strengthening the collaboration (between the participating laboratories): in a sense, you became an &quot;ambassador&quot; from your own institution to the host institution. A successful internship will be good for you, but it can also hold a lot of benefits for both institutions (that includes your supervisors) as if it goes well, they will be more likely to collaborate (more) in the future.
international experience. Where I am (France), people who plan to stay in academia and reach a permanent position are more than strongly encouraged to have an international experience. Basically, at least a few months internship abroad is a must for a permanent position, a year-long post-doc position or two abroad are much better. It is not a requirement, but always a person with international experience will get picked over the person with none.
both of these boil down to: post-doc. If is not unheard of that such an internship might actually be &quot;vetting&quot; the place for a post-doc. And this goes both ways: your host institution/professor sees if he likes working with you, and you see if you like it there.
specific benefit of the hosting lab. I have mentioned that there is many benefits of generally working in a different research environment and with different people for a while. But, specifically, there might be equipment, or test data, or an expert in the hosting lab that you just do not have in your institution.
diversity experience. If you have decided to stay in academia only on the basis of your PhD environment (as I have), you might not see the whole picture yet. Seeing different research labs, different environments, can only be a benefit.
publication While 3 months is not always enough to make a publication, you do want to get the most of the internship. Thus, most people get a publication out of it because they put a little bit extra effort (maybe not even consciously). And, your new publication will have a name of somebody other than your supervisor on it, which speaks to your ability to be able to work with different people successfully.

Now, of course, funding might be one obstacle. But other than that, if adequate funding is found, I see no reason for supervisors to object to such an idea.