How does the SpaceX Merlin engine control the mass flow rate to throttle its thrust?
Do the pintle injectors move?
Does the mass flow rate to the gas generator change?
SpaceX is not exactly forthcoming with detailed information about the Merlin's workings but we can figure some things out.
Tom Mueller, designer of the Merlin, gave a brief description of the engine in this video. He drew a sketch of the engine schematic which I have screen-captured for your use.
The only valves shown on this sketch were described by Mueller as (the LOX valve) "the valve that we use to turn it on and off" and (the fuel valve) as "the little valve". Not much help there.
However, spaceflight101.com has this image of the conceptual Merlin 2 engine. You will notice that it is remarkably similar to the Mueller sketch except with some more details.
From this, we can infer that the valves Mueller drew on his sketch were the Main Oxidizer Valve and the Main Fuel Valve.
This more detailed schematic shows gas generator control valves which would certainly be necessary to throttle the engine - increasing or decreasing the turbopump power to pump more or less propellant.
The remaining question is about the pintle injector, does it move? This writeup states that it moves, but only in the sense that it is spring loaded closed, and increasing pressure causes it to open more.
Merlin 1D is famous pintle engine from SpaceX. This pintle engine uses face shut off type pintle injector. Pintle is spring loaded and tightly pressed againist the pintle post. High pressure from LOX side pushes the pintle to open.
Putting all this together, I would infer that the Merlin throttles by controlling the propellant flow to the gas generator.