This map indicates that the Hittite Empire reached its greatest expansion c 1340 BC.
As to WHY an empire expands? I would submit that it is what empires do. In the case of the Hittites they ran into the countering Egyptian sphere of influence, fought a great battle to an apparent draw, made peace some years later and adhered to the treaty until they were overrun by the Sea Peoples. Both Egypt and the Hittites seem to have reached the point where they could project their power no further and reached stasis.
This has nothing to do with any fictional character named "Joshua" who was invented centuries later by a tribe of goat herders living in Judah.