Obviously it's not constant, but if it's uniform (enough) within the footprint of a single spot beam, the satellites can adjust for the global component both in their transmitter and receiver, and the mobile devices only have to compensate for (or tolerate) their local difference from that.
https://eudl.eu/pdf/10.1007/978-3-319-66628-0_41
doppler shift is not a constant for a stationary observer from Earth.