How were 20th-century mass population movements achieved?

For example, in Turkey/Greece after the Turkish war of Independence, my understanding is that millions of Muslims and Christians were shuffled around between the two nations. Or, after WWII, a ton of Germans were removed by the Soviet Union from Poland. Or, after the Algerian war for Independence a ton of pieds-noirs moved out of Algeria and back to France (although I think in this case the relocation was largely voluntary and happened over a longer time, but I could be wrong).

I'm sure these are all three very different cases, but they all involve hundreds of thousands, if not millions of people, being made to relocate substantial distances in the span of only a few years.

Where did these transferred populations end up? Who ended up in their old homes, or were thousands of homes left vacant? How were millions of people moved without turning it into a massive humanitarian crisis?