Welcome to the forum. Your post prompted me to think about our sink covers, and I am pretty sure they have been in the storage slot under the sink for a very long time! DW does not use them. Our sinks serve to hold last minute items we failed to put up in their proper storage spots when we pack up to leave…dog bowls, hand soap dispensers, a hand towel, roll of paper towels, etc.
To answer your question about the sink covers staying in place during travel…slow down!…and even then there are no guarantees! There is an overused phrase in this forum and others, but it is true…there is a category 6 earthquake going on to your fifth wheel while you drive down the cratered highways of these United States.