Many fled before of the end of the war because of the fear what soldiers of the USSR would do to them. Many of them who didn't fled were expelled, especially in Czechoslovakia, Poland and the new areas of the USSR. Those who weren't expelled had to deny their heritage and were often bullied, especially in Czechoslovakia because of the actions before 1938 of Germans there.

I am in no way an expert in this, but I do live in Belgium which is one of the countries that were given parts of Germany (Eupen-Malmedy). In today's Belgium, I believe there are around 75 000 German speakers. German is an official language in Belgium, these people have full Belgian citizenship and a great deal of autonomy (the German community have their own parliament). So, in short, for the Germans living in Belgium, not that much changed except for their nationality.

WWII:

By 1944, most of Eastern and Central Europe had been at war for 5 years. Millions had perished, cities had been destroyed, villages razed and locals slain. Blame was put almost unanimously on the Germans: they'd caused this pain and suffering, and the locals were mad as hell.

Article XII of the Potsdam Agreement, which was drafted and ratified after the war by the UK, the US and the USSR, called for the "orderly and humane" transfer of "German populations, or elements thereof, remaining in Poland, Czechoslovakia and Hungary". In total, aprox 12 million Germans and ethnic Germans were moved from Central and Eastern Europe back to Germany. How orderly and humane that transfer was is a point of contention for historians, with estimates ranging from 470 thousand to 2.5 million deaths - although places like the German Historical Museum's Center Against Expulsions put the number at around 600k.

The Potsdam Agreement regulated the expulsion of Germans, but they started as early as 1943, as the Red Army pushed back across Europe and German non-combatants fleed before it. Expulsions continued as those who stayed (or were left behind) in Poland, Czechoslovakia and Hungary were driven out by locals and national governments in "wild" expulsions. Some, like those who fled from Poland to Czechoslovakia, were then expelled for a second time.

After the Potsdam Agreement was put into force, the expulsion of Germans became more orderly, but not less awful. Some were put into internment or work camps (Poland, Yugoslavia, Czechoslovakia), others were just shipped out to Germany, some were "ethnically verified" and given local citizenship. Romania and Poland even allowed for German Minorities to exist. Interestingly, those minorities would often later use their minority status to gain the right to live and work in Germany, or even gain German citizenship. From 1950 to 2002 the number of Romanian citizens who identified as ethnic Germans fell from 400k to 60k - and this was before Romania joined the EU.

Poland's government made good use of the expulsions. Stalin had decided that Poland needed to give up its eastern territories to Lithuania, Belarus and Ukraine but would get quite a bit of German lands in return. (This whole issue and how German those lands were is a whole other thing.) Poles were moved from the eastern territories to the west, where they'd often literally move into abandoned German homes.

Did most stay or did most move to the new Germany territory?

Approximately 7.5 million Germans migrated or were expelled from Poland's "Recovered Territories" while a little over 1 million were given citizenship.

If they stayed, did they get to keep their property and receive citizenship from (for example) the Soviet Union?

German residents of Kaliningrad/Königsberg were expelled by force - this was the only part of pre-war Germany taken over by the Soviets. Those who took citizenship in Poland were allowed to keep their property.

If they moved, did they simply lose their property?

Yes. There's quite a bit of literature on how Poles who were forced out from their homes in the east took over formerly-German homes in the "Recovered Territories"