It was not uncommon for people to decide to start a new life. In the early years of the country, most people could easily pull up stakes and go into the “wilderness” (in the mid-1800s, that would be West of the Mississippi . . . which is why Illinois, Wisconsin, and surrounding states are called “the Midwest” - in the earliest years, that was as “west” as one would go and, when you consider Canada, those territories were in the middle of the western most areas). Someone wanted by the law would be able to start a new life (often with a new name) in this “new land.” This is why it is often hard to trace the pioneer settlers in Calif. when they came out in the late 1840s-1850s. Because a paper trail in the new area was essentially non-existent, finding documentation of these people may prove very difficult, at best.
On the same general topic, when people left “the old country” (meaning country “of origin” - in Europe, the British Isles, etc.), they were expected to bring a document from their ecclesiastical leader (priest, minister, etc.) to bring to America where they would produce it for the examination of the clergy in the new country. Without this paper, they would not necessarily be welcomed into the congregation (they might not be of good moral character). If we know the church they left and/or came to, that document may be available (or a record of its existence may be found in a church’s meeting minutes).