Uh... yes, evacuation is a thing. I imagine you will have heard of plenty of examples in the news, like whenever there is a natural disaster or there is a risk of regions becoming front lines in war.

The concept of evacuation from the countryside to a fortified place is probably as old as human settlement, but that's not really "new" land, so perhaps not what you're looking for. There are also plenty of examples of evacuation or relocation of communities threatened by enemy invasion. Generally speaking, since most human beings are not terribly effective fighters (due to age or illness or malnourishment or disability or simple lack of preparation), it has been a fairly common approach to get the hell out of wherever the enemy is headed, either collectively or while leaving behind the most physically fit to confront the threat.

From Greek history, there is the famous example of the Athenians, who evacuated their homeland in the face of the advancing Persians in 480 BC, leaving their emptied city to be razed to the ground. The Athenian population found temporary refuge on the island of Salamis and at the friendly town of Troizen in the Peloponnese. When it seemed like the Spartans were not committed to helping them get their homeland back, the male Athenian citizens threatened to simply put their families on board their warships and sail for Italy to start a new city.

The Athenians were not the only ones to respond to Persian expansion in this way. A few generations earlier, some of the Phokaians decided to leave Asia Minor and resettle on Corsica rather than face subjection. After the defeat of Xerxes, the Spartans realised that they would be unable to protect the newly liberated Ionian Greeks of Asia Minor against Persian retaliation, so they suggested that the entire Greek population of the area - the citizens of dozens of city-states - simply upped sticks and moved somewhere else, where they would be out of the Persians' reach.

As I said, this kind of thing is generally pretty common in history. The Greeks liked to argue that it is a community of citizens, not land or physical space, that makes a state; with the exception of the Athenians, nearly every Greek city-state's foundation story involved ancestors migrating from elsewhere. But it's not just common among the Greeks. The so-called "barbarian invasions" of the late Roman Empire should be understood similarly: entire Germanic and Gothic peoples moved west because other groups were pressing upon them from the east, making it impossible for them to stay where they were. Mass migration is subject to push factors as well as pull factors, and one of the strongest push factors are encroachment, invasion and war.