It depends heavily on where but for the most part to my knowledge people imprisoned for crimes the Nazis deemed non-political were left there to serve out their prison sentences.

There are groups of prisoner, especially in Germany who were deemed either professional or habitual criminals or who were due to the nature of their crimes deemed asocial and therefore transferred to Concentration Camps. Little research has been done into the history of those groups (mainly because they remained stigmatized and ostracized in post-war Europe) but this group included people connected to organized crime, people from poor backgrounds the criminalist paradigm at the time deemed "habitual" (i.e. poor people who in the opinion of leading criminalist had because of their family history "no other option" than to go on stealing e.g.) and so on.

In the occupied territories, prisons were often taken over by the German state security apparatus, i.e. the Gestapo. This meant their integration into the Nazi system of political persecution and brought with it a general change of regimes within the prisons most significantly felt in Eastern Europe.

In Poland for example Pawiak prison in Warsaw and Montelupich prison in Krakow were taken over by Gestapo after the invasion of Poland and immediately integrated into the Nazi system of persecution, meaning the became the place of imprisonment for the Polish inteeligentsia as well as street raids conducted by Nazi security officials. As for the prisoners already there, they were checked by the Gestapo for being political opponents and mostly left to serve out their sentences. However, on occasion, the Nazi authorities used the prisoners as hostages for reprisal actions. When German institutions or officials were attacked, there were reprisdal shootings and several of the prisoners both from Pawiak and Menteulpich prisons were shot as hostages.

In the Soviet Union, the situation differed. There prisoners were either checked as potential collaborators (if imprisoned by the authorities for anti-Bolshevist agitation or something similar) or in many cases summarily shot as was the case in Kiev and Minsk.

In other occupied territories were satellite regimes remained in place, the prison administration remained by and large in the hands of these governments (Belgium e.g.) and thus there was a semblance of continuity in the criminal system. However, keep in mind that German occupational authorities would often get involved because crimes such as black marketeering or theft took on a political nature in the course of the war.

Sources:

• Encyclopedia of the Holocaust, ed. Israel Gutman, vol. 4, New York, Macmillan Publishing Company, 1995.

• Stanisław Czerpak and Tadeusz Wroński, Ulica Pomorska 2: o krakowskim Gestapo i jego siedzibie w latach 1939–1945, Cracow, Muzeum Historii, 1972.

• Dieter Pohl: Die Herrschaft der Wehrmacht, Frankfurt a.M. 2002.

• Julia Hörath: Terrorinstrument der „Volksgemeinschaft“? KZ-Haft für „Asoziale“ und „Berufsverbrecher“ 1933 bis 1937/38. In: Zeitschrift für Geschichtswissenschaft. Bd. 60, H. 6, 2012, S. 513–532.

• Wolfgang Ayaß: Schwarze und grüne Winkel. Die nationalsozialistische Verfolgung von „Asozialen“ und „Kriminellen“ – ein Überblick über die Forschungsgeschichte. In: Herbert Diercks (Red.): Ausgegrenzt. „Asoziale“ und „Kriminelle“ im nationalsozialistischen Lagersystem. Beiträge zur Geschichte der nationalsozialistischen Verfolgung in Norddeutschland (= Beiträge zur Geschichte der nationalsozialistischen Verfolgung in Norddeutschland. Bd. 11). Edition Temmen, Bremen 2009.