If I'm interpreting the Wikipedia article correctly, the company did not move to France but had a French subsidiary. As you say, the French branch made aircraft engines during that period whereas the Spanish factory was requisitioned to make trucks for the Republicans during the Civil War period.
https://en.wikipedia.org/wiki/Hispano-Suiza
Interestingly, together with various other Spanish industrial companies, HS co-founded the company SEAT in 1940 and transferred its automobile activities to it. As such I guess any rights to Hispano Suiza automobiles are now vested in the Volkswagen group. The aeronautical activities continued separately beyond this date.