Highly redundant pathways often contain components whose functions are difficult to decipher from the responses induced by genetic or molecular perturbations. Here, we present a statistical approach that samples and registers events observed in images of intrinsic fluctuations in unperturbed cells to establish the functional hierarchy of events in systems with redundant pathways. We apply this approach to study the recruitment of actin assembly factors involved in the protrusion of the cell membrane. We find that the formin mDia1, along with nascent adhesion components, is recruited to the leading edge of the cell before protrusion onset, initiating linear growth of the lamellipodial network. Recruitment of Arp2/3, VASP, cofilin, and the formin mDia2 then promotes sustained exponential growth of the network. Experiments changing membrane tension suggest that Arp2/3 recruitment is mechano-responsive. These results indicate that cells adjust the overlapping contributions of multiple factors to actin filament assembly during protrusion on a ten-second timescale and in response to mechanical cues.