Hopefully you are using Moen Genuine Replacement parts and not Danco knockoffs; a huge difference in price as well.
Remove the cartridge from the valve with the water shut off at the main and use a shop vac hose covering over the housing of valve.
Have someone turn the water on at the main again and let the water come out of the valve. Both hot and cold will be running and watch for debri to come out of that hose into the tub.
If you notice a weak flow in this procedure.....a backwash will be necessary and it can be accomplished by using the cold into the hot by holding your finger over the valve with the hot water shut off at the tank and the hot side on the laundry tub faucet open.
The cold water from that valve will push whatever debri back through the hot side line and go to the largest opening in the hot, the laundry tub faucet. If you have an aerator on that LT faucet, remove it before doing the backwash.