I wouldn't worry that much about the road grime. Yeah, they'll pick up some junk along the way, but bikes can be cleaned. If you do end up getting hit with some snow/slush and road salt you could always stop at a spray wash and clean them off. Your bikes aren't going to suffer any long term damage from this. I personally wouldn't bother with covers, but they probably would help a bit.

As for security, this might be obvious, but I would not leave bikes in or on the car overnight, even locked up/covered. Definitely bring them into your hotel rooms. During the day in most cases it's probably fine to leave them on the car (assuming they're locked up) for short periods, like while you're in a restaurant or whatever. Assuming you're being smart about where you're parking, using a high quality U-lock, etc it'll be fine. The covers might help with this aspect as well.

Also might be worth contacting your auto insurance carrier to see if theft would be covered.