It depended quite heavily on what army said prisoners belonged to. For example, Jews who were part of the Red Army were usually summarily executed, others were sent to death camps such as Sobibor or Auschwitz. Outside of death camps, huge numbers of Soviet-Jewish POWs were murdered through slave labor, but so were their non-Jewish counterparts, as the Nazis generally did not follow the Geneva Convention when it came to Soviet POWs. Many of the prisoners involved in the 1943 Sobibor Uprising were Soviet-Jewish soldiers who had been captured and sent to Sobibor for slave labor and eventual murder. However, many British and American Jewish soldiers were protected by their fellow prisoners, or by their British/American citizenship which made the Germans more hesitant to break the Geneva Convention. However many Jewish Americans were held in much more brutal conditions than their caucasian counterparts (the Nazis did not view Jews as white). Sometimes they were executed, held in concentration camps, or tortured.
There is a similar dichotomy among Black soldiers. Black French or Black French Colonial soldiers in North Africa were routinely executed by the Germans, many never even being taken officially as POWs. However, because of their American Citizenship, most African American POWs were treated in accordance with the Geneva Convention, though more harshly. There were also cases of African American soldiers being summarily executed by the Nazis, but this was less common than among their Black French, or North African counterparts.
Most Hispanic soldiers were American and so were generally treated in accordance with the Geneva Convention, though of course there were many cases of abuse of all American POWs, especially minorities. The Nazis were generally unfamiliar with Hispanic Americans, but this didn't always mean they were treated like their white counterparts. It was a Latino soldier, Corporal Anthony Acevedo who cataloged the abuses, including starvation, forced labor, death marches, and rape, against himself and other American POWS (mostly Jews), as well as all the deaths which occurred in the Berga concentration camp.
Muslim, East Asian, Southeast Asian, and Indigenous Americans fought in the British, Canadian, Anzac, and American forces. The Nazis had less specific racial ideology surrounding Asians and Middle Easterners, so these soldiers were likely to be treated differently. More research needs to be done on the experience of these groups, but they were likely treated more similarly to their counterparts than Black or Jewish POWS.
Works Cited
"Escape From Sobibor" by Richard Rashke
https://encyclopedia.ushmm.org/content/en/article/blacks-during-the-holocaust-era
Yad Vashem, Shoah Resource Center, "Prisoners of War"