Cilia and flagella are microtubule-based organelles that protrude from the cell body. Ciliary assembly requires intraflagellar transport (IFT), a motile system that delivers cargo from the cell body to the flagellar tip for assembly. The process controlling injections of IFT proteins into the flagellar compartment is, therefore, crucial to ciliogenesis. Extensive biochemical and genetic analyses have determined the molecular machinery of IFT, but these studies do not explain what regulates IFT injection rate. Here, we provide evidence that IFT injections result from avalanche-like releases of accumulated IFT material at the flagellar base and that the key regulated feature of length control is the recruitment of IFT material to the flagellar base. We used total internal reflection fluorescence microscopy of IFT proteins in live cells to quantify the size and frequency of injections over time. The injection dynamics reveal a power-law tailed distribution of injection event sizes and a negative correlation between injection size and frequency, as well as rich behaviors such as quasiperiodicity, bursting, and long-memory effects tied to the size of the localized load of IFT material awaiting injection at the flagellar base, collectively indicating that IFT injection dynamics result from avalanche-like behavior. Computational models based on avalanching recapitulate observed IFT dynamics, and we further show that the flagellar Ras-related nuclear protein (Ran) guanosine 5'-triphosphate (GTP) gradient can in theory act as a flagellar length sensor to regulate this localized accumulation of IFT. These results demonstrate that a self-organizing, physical mechanism can control a biochemically complex intracellular transport pathway.