Tidal effects on Earth...a couple of cents worth
It seems logical that lunar gravity would have at least some tidal effect on the solid and hot liquid parts of the Earth. Obviously, there is measurable movement of the crust due to tidal effects.
There are organizations which study these tides.
As Dan and others have said, how much such tidal forces contribute to movement of the mantle inside the earth is problematical. However, such tidal pulls have been correlated with volcanic eruptions.