A Dispatcher reported an aircraft deviated from their ramp taxi clearance then had to stop suddenly to avoid an aircraft being pushed back from a gate.

Narrative

Aircraft Y was pushed off XX; while Aircraft X [Gate] XY already had the conditional to push when other traffic behind them taxied out the north. Aircraft Y called for taxi and was first instructed to start [Line] 1 north with me. Abeam [Gate] XZ; Aircraft Y was instructed to transition 2 line to 3 and contact metering; and read it back correctly. I began attending to other traffic that was calling about gates and entering 4. Aircraft Y failed to transition to 2 as instructed and can be seen hitting the brakes at [Gate] XA because XY was pushing back. Thankfully Ramp noticed Aircraft Y approaching and stopped the push; than began pulling them back into the gate.767 notified me of the issue but I never heard from Aircraft Y again.Additionally; it appears Aircraft Y began turning wide towards 2 line to continue taxiing around the tail of 767 before they were pulled back into the gate.They remained 1 line to 5 and eventually taxied out of the alley.