Current estimates of climate-driven human migration often rely on relationships between population and a mean climate state. However, compared to other environmental impacts such as crop failure or flooding, human migration measurements are often only available on longer timescales, requiring metrics of climate that can capture both long-term trends and changes in variability on the same timescales. Using a gridded global data set, we estimate internal migration responses from a combination of trend and variability climate metrics to understand how different characterizations of environmental change influence human migration within countries. We compute monthly and annual metrics using end-of-20th-century temperature and precipitation observations to derive empirical relationships between environmental change and migration between 1970-2000, which are heterogeneous by location characteristics. We then apply the historical relationships to project future global migration and climate metrics calculated from the CMIP6 models through 2050. Using a machine learning approach, our results suggest that historical estimates of climate-driven migration depend on both climate and climate change. We find that the influence of changes in baseline climate state and transient metrics of climate change can have conflicting effects on internal migration and that motivation to migrate may be hindered by income constraints. As the climate shifts towards a forced system, we can expect regional heterogeneity in the type of environmental changes that have driven human migration in the past to increase. Finally, we compile micro-datasets on migration to estimate how increases in internal migration would lead to international migration pressures around the world.