Does any of the apps like Donkey Republic or Nextbike allow us to rent a cycle in one city and return in another? I read the instructions, but it is not clear whether to drop-off-location can be in another city.
Nextbike operates bike shares in multiple locations in Europe.
For Germany, the Nextbike.de information page says:
Bikes can only be returned in the same city except for: KVV.nextbike, RVK e-Bike, Bergisches e-Bike, RSVG-Bike, Eifel e-Bike, metropolradruhr and VRNnextbike. Information on whether bikes can only be returned at fixed stations or also flexibly within a Flexzone can be found on the website of the respective city. Find your city in the location search.
In Czechia, NextbikeCzech FAQ page says:
What is the "Flex Zone"?
The flex zone is an extended area into which the bike may be returned with impunity and thus passes as an extended official station. Flex zones are only in certain cities and are marked on the map with blue highlights.
There are other Nextbike sites for other countries (UK, Croatia, Bosnia and Herzegovina, ...), which may have different rules, so you should check the Nextbike site for your country.