I have been reading Hans Ulrich Rudel's "Stuka Pilot," and he mentions that his wing was rebased out of the Cauldron after the 6th Army was surrounded at Stalingrad. They were rebased to a nearby airfield behind German lines, but how would they have evacuated their ground crews? Did they have cargo planes attached to the unit? Rudel doesn't give the impression that they left their ground personnel behind.
There are several aspects to this question. First as the Luftwaffe was organized, the maintenance crews were not under the command of the squadron, but rather by a ground district command unit (luftgaue). Basically it was this unit's responsibility to provide ground crews. It was an interesting organization since whole squadrons could be deployed by only flying to their new airfield, no strings attached.
The other facet of this was of course the airlift that the Luftwaffe tried to facilitate, while being a failure still succeeded to evacuate thousands of wounded soldiers and some Luftwaffe personnel out of the city. The Luftwaffe did transfer air units in and out of the city especially early in the campaign, right before the soviets raised an aerial blockade of the city, that didn't stop the airlift, but did slow it down.