Intercellular propagation of protein aggregation is emerging as a key mechanism in the progression of several neurodegenerative diseases, including Alzheimer's disease and frontotemporal dementia (FTD). However, we lack a systematic understanding of the cellular pathways controlling prion-like propagation of aggregation. To uncover such pathways, here we performed CRISPR interference (CRISPRi) screens in a human cell-based model of propagation of tau aggregation monitored by FRET. Our screens uncovered that knockdown of several components of the endosomal sorting complexes required for transport (ESCRT) machinery, including charged multivesicular body protein 6 (CHMP6), or CHMP2A in combination with CHMP2B (whose gene is linked to familial FTD), promote propagation of tau aggregation. We found that knocking down the genes encoding these proteins also causes damage to endolysosomal membranes, consistent with a role for the ESCRT pathway in endolysosomal membrane repair. Leakiness of the endolysosomal compartment significantly enhanced prion-like propagation of tau aggregation, likely by making tau seeds more available to pools of cytoplasmic tau. Together, these findings suggest that endolysosomal escape is a critical step in tau propagation in neurodegenerative diseases.