You wouldn't ever see "the climate changed so we moved" as a cause here, because climate change takes a few generations to fully manifest. People without written records and good observation tools wouldn't be able to spot or record the climate change. But such people would certainly notice when agricultural output drops and decide to move to places where the situation is better.
Such wholesale migrations have been the cause of many conflicts. Steppe nomads have migrated to Europe on many occasions over the last two thousand years and each time it has led to war. Even within Europe this sort of migration was observed. For example, Julius Caesar fought Germanic tribes who had decided to migrate to Gaul because the grass was greener on that side of the Rhine.
Populations regularly outgrow their food supply and migrate. That isn't climate change. They also exhaust their land with poor farming practices, that isn't climate change, either.
Such wholesale migrations have been the cause of many conflicts. Steppe nomads have migrated to Europe on many occasions over the last two thousand years and each time it has led to war. Even within Europe this sort of migration was observed. For example, Julius Caesar fought Germanic tribes who had decided to migrate to Gaul because the grass was greener on that side of the Rhine.