A coarse-grained computational model is used to investigate the effect of a fluctuating fluid membrane on the dynamics of patchy-particle assembly into virus capsid-like cores. Results from simulations for a broad range of parameters are presented showing the effect of varying interaction strength, membrane stiffness, and Membrane viscosity. Furthermore, the effect of hydrodynamic interactions is investigated. Attraction to membrane may;promote assembly, including for subunit interaction strengths for which it does not occur in the bulk and may also decrease single-assembly time. The membrane budding rate is strongly increased by hydrodynamic interactions. The membrane deformation rate is important in determining the finite-time yield. Higher rates may decrease the entropic penalty for assembly and help guide subunits toward each other but may also block partial cores from being completed. For increasing. subunit interaction strength, three regimes with different effects of the membrane are identified.