The mass of a star is assembled through a converging motion in the large-scale turbulent flow. Because compressions in the supersonic turbulence results in a filamentary structure of the dense gas (from intersection of postshock sheets), the converging motions feeeding an accreting star occur predominantly through dense filaments.
We have suggested that this filamentary inflow not only controls the main growth process of protostars (hence their accretion luminosity), but also much of the pre-main-sequence (PMS) accretion for most stars (see Padoan et al. 2005, ApJ 622, L61).
In this picture, the circumstelar disk acts as a buffer between the large-scale mass inflow and the star, modulating the inflow rate into an actual accretion to the stellar surface.
The final phases of the PMS accretion are probably dominated by a Bondi-Hoyle type of accretion (see Padoan et al. 2014, ApJ 797, 32), with accretion rates much too low to increase significantly the final stellar mass.