Which year is your Mustang, and how many miles does it have? Where is the coolant temperature gauge needle? All the way to the H mark, or just beyond the middle?
So your Mustang was working fine before the gear swap? The only thing related to a gear swap I can think of is the higher engine load under acceleration, and the higher RPM, after gear swap. They shouldn't touch your engine for this work.
Is your coolant level good? Did you check to see if there is any engine code? I wonder if your thermostat is stuck closed.
2014 Mustang V6 3.7L 6-spd manual, with MGW shifter, Airaid CAI, BBK shorty headers, Corsa Sport axle back, strut tower brace, JLT oil separator, GT500 differential cover, BG Synchro Shift II gear oil, Redline differential oil.