The problem tends to not be with the tremolo but rather the nut or the tuning machines, either that or the strings are not stretched out.
If the strings are not stretched out right, they are either going to go flat as the strings stretch out in time, and if they are going sharp when using the trem arm, then it's either they are binding in the nut (slots too thin), or are so stretched out they are relaxing back at a happy medium where they should be.
I rarely find it's the tuners unless the winding around the capstans at the headstock is bad. If the strings are overlapping a lot, and the winding around the tuning pegs is just a big random mess, then it's going to go sharp when the string snags on a higher spot of string, and go flat when tension is released when the string settles into a deeper spot.
The only time it's an issue with the bridge is when the pivots are VERY badly mushroomed out, and then the guitar seems to still be in tune, just a hair flatter if you push down on the bar, or a hair sharp if you pull up...I've never encountered that with a Fender Dynamic Vibrato though....just Floyd Roses and twin pivot Stratocaster trems.
