In the past year, colleagues or collaborators have informally extended an invitation to visit their University, maybe to meet potential collaborators and give a talk. In general, is it normal for the "host" institution to provide any support? Does one's "home" institution normally provide support to do so?
Note: I'm currently a graduate student and am primarily asking for potential opportunities.
If you have been invited to visit, the default assumption in my realm is that the inviter will cover expenses unless they tell you up front that they don't/may not have money to fund all or any of your visit.
Usually the thing to do when you want to take an advantage of an open, informal invitation is contact the person, reminding them of your conversation, and ask about or suggest times to visit. If their reply does not make it clear if they intend to fund your visit, just ask them.
informally extended an invitation ... is it normal for the "host" institution to provide any support
The keyword here is "informally". For a formal invitation (to give a colloquium, to interview, ...) it is indeed customary that the host institution pays unless agreed otherwise. For an informal invitation among colleagues, you should not assume anything. Just ask, but be prepared that the answer may easily be that they don't have money or are not willing to pay for your visit.
Does one's "home" institution normally provide support to do so?
This depends entirely on your home institution, its budget situation, and as how valuable your visit is perceived by your home institution. Here in Switzerland, many institutions would pretty much auto-approve any such business visits as long as they are not horrendously expensive. In Austria, it was sometimes hard to get visits approved.
I agree with Kimball's answer. If I would like that somebody visits me but don't want to fund the visit, I would tell at the very beginning that I do not have funding for the visit. But then I would not call that an "invitation".
Sometimes people say something like "It would be great if you could visit our group when you are around." I would not call this an invitation but an "offer for deeper collaboration". If you've got some "invitation" like this and follow up on this by email, I would suggest to write something like "I would like visit and also could prepare a talk, but unfortunately, I do not have any funding available." (if this is true), and then see what reply you get.
If it is clear that the hosting group will not fund you, but you really want to go, then the first person to ask about funding is your advisor. More advice is not possible in general since universities/faculties/departments/groups differ a lot in this respect. Sometimes the department may have some travel money for which you may apply (but often this is not the case), sometimes there are other resources available, but a general advice is impossible.
