Epistatic interactions can make the outcomes of evolution unpredictable, but no comprehensive data are available on the extent and temporal dynamics of changes in the effects of mutations as protein sequences evolve. Here, we use phylogenetic deep mutational scanning to measure the functional effect of every possible amino acid mutation in a series of ancestral and extant steroid receptor DNA binding domains. Across 700 million years of evolution, epistatic interactions caused the effects of most mutations to become decorrelated from their initial effects and their windows of evolutionary accessibility to open and close transiently. Most effects changed gradually and without bias at rates that were largely constant across time, indicating a neutral process caused by many weak epistatic interactions. Our findings show that protein sequences drift inexorably into contingency and unpredictability, but that the process is statistically predictable, given sufficient phylogenetic and experimental data.