It's been a couple years since I was going through the application process, but I found sites to be very accommodating. I agree that you probably don't want to cancel or move an interview you've already scheduled, but if you let the new site know that you're already booked on the available dates, they might let you come in on another day. I even had one site add an interview day because so many people were booked up by the time they sent out their invitations. If all else fails, you can see if somebody would be willing to trade with you. In the end, if a site isn't understanding about all of this craziness, you might not want to be there.