Actin polymerization in vivo is regulated spatially and temporally by a web of signaling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodeling actin meshes and bundles. In particular, we investigated how molecular motors regulate growth dynamics of elongated organelles of living cells. Our simulations show that some processes, such as binding and unbinding of capping proteins, may be dominated by rare events, where stochastic treatment of filament growth dynamics is obligatory. We also studied mechanical regulation of the growth dynamics of lamellipodia-like branched actin networks. In such networks, the treadmilling process leads to a concentration gradient of G-actin, thus G-actin transport is essential to effective actin network assembly. We shed light on how actin transport due to diffusion and facilitated transport such as advective flow and active transport, tunes the growth dynamics of the branched actin network. Our work demonstrates the role of molecular transport in determining the shapes of the commonly observed force-velocity curves.