We aim at insight into the unsteady kinetics and the formation of spatiotemporal patterns of steps during the crystal growth in systems, in which the growth rate is controlled by the rate of supply of material. For this, we apply phase-shifting interferometry to the crystallization of the protein ferritin. We find that the locally measured growth rate, step density and step velocity fluctuate by up to 80-100% of their average values. The fluctuations are due to passage of step bunches generated at the facet edges due to unsteady surface nucleation. The fluctuation amplitudes decrease with higher supersaturation and larger crystal size, as well as with increasing distance from the step sources, even while the average value of local slope, a destabilizing factor, increases. Since size and supersaturation are parameters affecting the solute supply field, we conclude that fluctuations are rooted in the coupling of the interfacial processes of growth to the bulk transport in the solution. To understand. the counterintuitive suppression of the instability, we analyzed the step velocity dependence on local slope and found only a very weak interaction between the steps, likely due to competition for supply from the solution. Accordingly, the step bunches propagate with the same velocity as elementary steps. We conclude that in transport-controlled systems with noninteracting or weakly interacting steps the stable growth mode is that via equidistant step trains, and randomly arising step bunches decay. Stronger step interactions may reverse this conclusion, or slow the rate, at which step bunches decay and stability is reached.