How to invite potential collaborators for personal visit? No starting point for scientific collaboration is better than face-to-face conversation. Conferences are good places to meet potential collaborator, but another way is to invite them to visit your institute/group and negotiate for possible collaborations.

However, when inviting a colleague from long distance (e.g. crossing the Atlantic), it is difficult to cover the cost of the flight (particularly as it significantly varies and hard to justify its necessity). Sometimes, the visitors has better access to available fund to cover his/her trip. Of course, it is easier to cover the cost of accommodation.

How do you invite a potential collaborator to visit your institute when you cannot cover his/her flight cost? Isn't it rude to invite someone and quote you must come on your own expenses as we cannot cover it? Note that the relationship is quite formal with least personal acquaintance (no joint work yet).
With apologies, and a firm understanding that lack of funding may be a dealbreaker.

It would certainly help if you can cover part of the expenses: "If you are ever in [Country X], I hope you will also visit [my institution] and we would be happy to pay for your hotel [or else do you have a couch?] and pay for around [small number] towards your travel expenses. You are of course very welcome to make a special trip but I regret that funding is tight and we would be unable to pay your international travel expenses."

... or something like that.