It varied from case to case -- there was no consistent or unified response by native cultures. The western hemisphere was a fairly violent and culturally divided place before the arrival of Europeans, with low-intensity conflicts occurring nearly continuously. Large dominant cultures like the Incas and Aztecs were not the norm, and these both controlled only a small fraction of the territory of the 'new world'.

The Spanish conquistadors took advantage of the disunity between cultures and tribes and often used native allies as they warred against one or another native culture or tribe. The same can generally be said of other colonial powers. Of the colonial powers the French tended to have the most diplomatic approach, but perhaps only because of other more urgent strategic concerns in Europe during this era.

In some cases as peoples fled their lands, they conquered the lands of others to create new homes. In other cases they were absorbed into another culture or tribe. In others, they were slaughtered as they encroached.

Just depends. No single dynamic describes it all.