In a cluster crystal, each lattice site is occupied by multiple soft-core particles. As the number density is increased at zero temperature, a " cascade" of isostructural phase transitions can occur between states whose site occupancy differs by unity. For low but finite temperature, each of these transitions terminates in a critical point. Using tailored Monte Carlo simulation techniques, we have studied such demixing cascades in systems of soft particles interacting via potentials of the generalized exponential form u(r) = ε exp[-(r/σn )]. We have estimated the critical parameters of the first few transitions in the cascade as a function of the softness parameter n. The critical temperature and pressure exhibit non-monotonic behavior as n is varied, although the critical chemical potential remains monotonic. The trends for the pressure and chemical potential are confirmed by cell model calculations at zero temperature. As n → 2+, all the transitions that we have observed are preempted by melting although we cannot rule out that clustering transitions survive at high density.