I have initially posted this question in relation to the Romanian army, which was second to the German on the Eastern front, but I have updated the title to include all other non-German armies on that front.
On Wikipedia I see an article on Romanian prisoners of war in the Soviet Union, but not one on the reversed situation.
I find the same situation when searching on Google "Soviet prisoners in Hungary".
I know the treatment of Soviet prisoners by Germany was very harsh, amounting to extermination.
I know Romanian forces participated in the initial success of Germany on the eastern front and took some prisoners (like those mentioned here, here, here, and in other articles on the same website, here, etc).
Somewhat vaguely, from my own family I heard about some eye-witness (or rather actor) of events where Romanian soldiers had been ordered to summarily execute Soviet prisoners.
Information on Romanian army being involved in mistreatment and killing of civilians - especially Jewish - is easily accessible (like here or here) - drawing a trend that might have also included other Soviet population and prisoners - but information on Soviet prisoners taken by Romanian army specifically is not easy to find.
Were Soviet prisoners taken by Romanian troops immediately or later put under German custody? (Maybe related to Was Romania under German command during WWII?.) or did they remain under Romanian custody? Were they sent to Germany (and Poland) camps, camps in Romania or simply behind front lines?
What about POWs taken by Hungarians?
As for other Axis members, I imagine Italians (and the French legion) were far enough from their own territories to leave any such task to the Germans, while Bulgarian army didn't took part to the attack on USSR.