What do you mean by "chain drop"? does the chain actually come off the chainrings completely or does the chain derail back to the inner ring? Does the chain make it all the way up onto the middle ring at all?

The first thing I'd look at is the ability for the ft der. to move far enough to have the chain lift completely on to the middle ring. You could place the chain on the middle ring by hand then see if the der cage is centered on the chain. A common problem is the cable tension is not quite enough to have the lever move the cage enough. Shimano ft ders are sensitive to cable tension in a variety of ways, this is but one of them. One of the complaints that we get at the shop is that the ft der doesn't shift well after a while, like after warming up things change. Well the only thing that changes is the rider's awareness. If the lever is not thrown to a full amount, as can happen after a rider tires, then the shift might not be completed.

Of course making sure there's no bent or twisted links or teeth is good advice. Worn chains get more side to side play and the need for cage movement increases over time, to compensate for this added slop.

While condition and tune condition are factors in ft shifting there is a LARGE technique aspect and it is the largest variable. Andy.