"people with migration aspirations" sounds awkward in my ears. Not wrong, but awkward. More "normal" English formulation would be "would-be (e)migrants", or your less colloquial "people wishing to (e)migrate" ).

In my non-DMS ears, "Menschen mit Migrationsbestrebungen" sounds like a deliberate echo of "Menschen mit Migrationshintergrund" - do you think that is intended here? Does the text relate to Germany as the destination country of choice? If not, I would not see a necessity to retain a parallel structure - "persons with migration background" is not nearly used as much internationally as it is in Germany, and its inclusion of second and third generation immigrants seems irrelevant with regard to help being offered to people in their countries of origin (if that is indeed what this text is about).

Hmmm ... richtet sich das an - möglicherweise gut ausgebildete - Fachkräfte, die in eines der DACh-Länder einwandern oder eher an Deutschmuttersprachler, die auswandern wollen ? ...

FYI vocation, "Flight background" would be interpreted by most English speakers to mean: those who had served in the airforce :) The English would probably be 'refugees'.

Also, generally, English does not use People with x-background in the way that German does. Here, you could say something like "We will help those who want to immigrate [to GER] to find a job which is fairly paid, which improves their standard of living, etc... " or "We will help aspiring immigrants [to GER] to find a job ..." or "We help people who want to immigrate to Germany find a job which is well-paid and will improve their standard of living."

Thanks a lot, procratinator and Lonelobo. Very appreciated and helpful answers. Yes, the text is exactly about helping aspiring would-be immigrants - possiby refugees - to find a decent and fairly-paid job.