wheel tims

I have now fitted radial tyres to my 59 skyliner.Wow .what a difference they make.I can now drive it like I stole it .However, I am having a problem with my wheel trims.One rear wants to keep coming off and one front actually appears to move around the wheel.I can fit the trim with the valve through the hole.After a journey the valve has dissapeard out of sight.What do I need to do to stop this happening.I guess there is now a small amount of flex in the wheels due to the stiffer sidewalls.Any help appreciated

I had the same problem with my 1950 Plymouth, all 4 hub caps were doing the same thing, I put in metal valve stems, that STOPPED it, lOL