Is it true that after the Assyrians captured a city they allowed the conquered people to remain in their homeland

FALSE. The Assyrians had a consistent policy of rearranging populations. In the Ancient Period, people were typically connected quite strongly to the land they lived on, often to such an extent that simply deporting them to a different land would result in their identity shifting and assimilating into a new group. The Assyrians took advantage of this by forcibly rearranging populations. This had the effect of limiting resistance to Assyrian occupation and the weakening of Non-Assyrian culture.