The canine transmissible venereal tumor (CTVT) is a cancer lineage that arose several millennia ago and survives by "metastasizing" between hosts through cell transfer. The somatic mutations in this cancer record its phylogeography and evolutionary history. We constructed a time-resolved phylogeny from 546 CTVT exomes and describe the lineage's worldwide expansion. Examining variation in mutational exposure, we identify a highly context-specific mutational process that operated early in the cancer's evolution but subsequently vanished, correlate ultraviolet-light mutagenesis with tumor latitude, and describe tumors with heritable hyperactivity of an endogenous mutational process. CTVT displays little evidence of ongoing positive selection, and negative selection is detectable only in essential genes. We illustrate how long-lived clonal organisms capture changing mutagenic environments, and reveal that neutral genetic drift is the dominant feature of long-term cancer evolution.