The bridge itself should not move. What could happen is the trem springs loosing tension, which would result in a bridge that is floating (or floating too much) and would somewhat move the bridge forward. Also this would make tuning the guitar a nightmare, as any change of tension on the strings would affect how much the bridge floats.
If this is the case such guitars usually have ways to tighten the springs or to add more springs.
This sort of thing should be controllable by action and intonation though. If the frets themselves are out of tune there are other possibilities:
- Fret wear will affect the contact point of the fret and thus eventually affect intonation
- Bends or twists in the neck might affect the profile of the distance to the strings, which will affect intonation
- An offset nut will mess up the whole intonation of the instrument
So try to check for:
- Is the bridges sitting tight on the body or is it floating more than it should be?
- Are the frets looking fine?
- Can you spot any sorts of unusual bends or twists on the neck?
- For the nut check the intonation with a capo at the first fret (if the nut is not positioned correctly this should still intonate well).
EDIT: Another point could be action. As far as I can see from the image you’ve got your bridge action set up rather high. The higher your action is the more the string tension will increase upon fingering a note (increasing the pitch). This means that with a high action you need to intonate your frets lower to amend this. This can also affect the general intonation of your frets. So you might also check if the action is rather high.