As far as the time on the back of the car driving, I would not give it a second thought. I have done many cross trips with road bikes and MTBs on the back of my car for 4-6K miles. Yes, the bikes can get dirty but it does not hurt anything. Just be sure the chains are lubed.

I would think twice about doing it in the winter if going somewhere they use salt, though.

Security is a different matter, though. And having two different bikes means you are always leaving one unattended when riding. Its doable (I’ve done it) but takes some planning and extra steps.

As far as locking up, I have a long length of massive hardened steel chain with a 29er tube slipped over it that goes on once on the road. I also have a couple of u-locks that get tossed on if I am particularly worried.