FR type trems are a bitch to tune and historically are a pain to keep in tune. The most important thing is to ensure that the strings are fully stretched to begin with. The root cause of the problem is that the tension in the tremolo springs is supposed to balance the tension in all 6 strings, so if one string stretches, the whole tremolo moves to equalise the tensions again, resulting in 5 sharp strings. Beyond that issue, the pivot is a 'knife' edge that is supposed to remain in a stationary point on the posts. In reality, cheaper FR-type trems have knife edges that aren't very hard so they can become rounded with use, resulting in them being able to 'roll' off of that pivot point. Change the pivot point, change the tuning...
This problem is also seen due to the tremelo post itself shifting. Setting the post height sets the action height, so they are nearly always adjustable, screwed into thread inserts set into the body. The metal-metal tolerances and vibration can allow the post to effectively move in 3 dimensions, enough to shift the tuning. Better trems have set screws to lock the posts to the inserts to avoid this from happening.
Of course, gross incompetance should never be ruled out either

Having said all that, I wouldn't recommend a FR-type tremolo for dropped tunings since you'd have to change the trem spring tension every time you change tunings (unless of course you can dedicate one guitar to each tuning!) FR trems are also useless for playing double stop bends since sharpening one string flattens the others.

Ibanez have recently introduced a new tremolo, the Zero Resistance tremolo which addresses most if not all of these issues. It uses ball bearings instead of the knife edge, and has some gadget that will keep the guitar in tune through a G/B/high E string break, and also allows double stop bending.
(It's included on the 2003 onwards S470 guitar I mentioned earlier)