Mk3 Escorts have always been prone to this, particularly on the back doors for some reason, but you don't need to worry about those

It's unlikely that your car has the original system still fitted as mk3 escorts didn't have a solenoid in the drivers door - just a switch. This meant that if you fitted remote locking the drivers door wouldn't work.

Someone has probably got around this by fitting motors from a Mk4 Escort, and this is probably out of adjustment as they take a bit of fiddling with. Take off the door panel and have a look at the rod that connects the motor to the latch, it probably needs shortening slightly and bending it slightly might be sufficient.