It seems the Senate and Patrician level families had more to lose if a tyrant came to power while the majority of people might not be effected or possibly could even benefit. Especially when it came to people like Julius Caesar who, at least on the surface, seemed much more populist than many in the Senate.
Interesting question. I think you'd find that since Roman society was so centered on networks of patronage, many people had a vested interest in the power/influence/wealth of the individuals who would be diminished by tyrants. If Caesar and his followers are grabbing all the goodies, Pompey's retainers feel the loss as much as Pompey.