If they have interviews in mid-July, they will make decisions in late July at the earliest. Give a week or two to negotiate the details and sign a contract. You're then in a position where a candidate doesn't even have sufficient time to give their two-weeks' notice as is common in the US (with much more time required in other countries if you want to quit your job). You also have to find a place to live and actually move.
This will be no news to your new department. They understand these timeline issues, and I cannot see anything wrong in asking them for their timelines. In particular, I cannot come up with any reason why such a question could possibly be construed in any negative way: after all, it shows that you are serious and are thinking through the practicalities of making the commitment to move. This is better than having a candidate who never seems to care about these issues, possibly because they have no intention of accepting the offer anyway and only wants to use it for negotiations at home.
In other words: Ask whatever questions you have when you're there. They invited you, so they at least think about wanting you to come. They will be more than happy to talk you through the things that aren't clear to you, and they will likely also listen to issues that will present hurdles to you (such as giving your current employer the legally required minimal time for terminating your employment, moving in time, getting ready for classes while moving, etc). None of these questions should catch them off guard.