Mine in the rear (WeatherTech) flap all around with the doors off but is tucked pretty under the seat/hump so it holds it in even if it is flapping all around.
However I went out on the freeway and 75mph is a bit too much and I was about to lose the mat out the back but pinned it on the rear seat and one armed it until the exit ramp.....
Sadly, I didn't read this sooner. One of my rear carpet floor mats did indeed fly out with the doors off and I didn't notice until I got home. So now I need to figure out how to buy another one or if I can get a new set. I have the four door with the washout floors.
same - one of mine flew out on the way to work today (the other I saw raising up and reached back and grabbed it just in time) - retraced my steps and couldn't find it.
Because I felt bad leaving my dog behind, I've mostly been keeping the back doors on when going doors off. I've found the @Mabett rear cup holder has been pretty helpful in keeping the rear floor mats a little more secured when I take the rear doors off. That being said, if I'm going more than 45 MPH, the rear mats come off and get stored in the trunk.