To say that Native Americans migrated would almost imply that they chose to move to new lands for personal gain, when in actuality, they were forced to migrate when the government forced them onto reservations. The stories of various tribes and the trials and tribulations they faced during all of this turmoil are heart-breaking and leave a very dark mark on American history.
Gold was a tremendous lure for many Americans seeking to get rich quick, and Oregon and California were states to which many migrated. There was also cheap land available in places like Oregon where one could start anew. Because there were so many men in the mining towns, women often migrated there is search of husbands.
I have to support the answer which denotes the move west to coincide with financial reasons. People seemed to be drawn by the promises of riches and success the West offered. Land was offered at little or no cost and the promise of a better life was too promising to pass up.
Native Americans did not migrate "westward to Mexico, California, and Oregon after 1815." The original Native Americans migrated from Asia over the landbridge between what is now Russia and what is now Alaska when the Bering Strait was dry land. From that entry point, they scattered and spread throughout the North American continent.
The typical answer to this is that life was much harder for white women than for white men during the westward migration. They had to try to keep their families fed and clothed in tough conditions. They might become pregnant or have to give birth in the same conditions. Women in the West were generally given more responsibility than those in the East, which is one reason they would get the right to vote earlier.
People migrated for different reasons, but primarily for economic motives. People moved into Mexican territory because the Mexican government was offering cheap land, though the issue of slavery (which was outlawed in Mexico when they gained independence from Spain) quickly became an issue, as settlers wanted to raise cotton there. This, in fact, was one of the major issues that led to the Texas Revolution. As for California and Oregon, the bulk of migration took place immediately before and after the Mexican War, with the peak being the years of the gold rush in California. Some people went to Oregon for very different reasons, as the example of Marcus and Narcissa Whitman demonstrates. They went to proselytize to Indians in the Northwest. But by far, most people went west for increased economic opportunity.