It bonds to craft foam (the cheap stuff from Michael's art store) like glue.
It sticks to the smooth side of the floor mats about as well as PVA does if you do it badly - it's possible to tear it off, but it makes a bit of a mess.
It bonds to the sanded EVA nearly as well as craft foam - chunks of foam come off if you rip it out.
It doesn't stick to the textured side AT ALL - comes off like it was poured over wax.
That said, it MIGHT work a lot better with glue down first. But I suspect the textured side is the more treated, resistant side - it seems less porous if you really look. Probably to make it easier to clean spills and resist wear in your garage or whatever (that IS what those mats are actually for). So I'm not sure how well glue would soak in for bonding purposes.
Of course, you'll loose SOME detail - you're spreading another layer on top! But the texture doesn't disappear entirely, each little raised "island" just gets rounded out.