Scandinavian church records in the US can jump-start your research.
When they settled in the US, Scandinavian immigrants founded churches and other religious institutions—mostly but not entirely Lutheran. By the turn of the century, for example, the Danish Lutheran church had established a theological school in Wisconsin, an immigrant mission in New York, and an orphanage in Chicago, and numbered 56 ministers. Each country introduced its own church body to the United States, so the Danes founded what became the American Evangelical Lutheran Church (AELC). That merged with the ethnically Swedish (among others) Lutheran Church in America (LCA) in 1961. The LCA in turn merged with the American Lutheran Church (ALC), which had previously incorporated the Norwegian Lutheran Church of America (NLCA), in 1988 to form the Evangelical Lutheran Church in America (ELCA).
These bewilderingly similar-sounding Lutheran churches are worth noting because in the early immigration years many kept up the assiduous record-keeping their congregants had been subject to back home. In addition to baptisms, marriages, and burials, some recorded comings and goings much as their state-sponsored Scandinavian counterparts did. You might be able to find answers about your ancestors in these American records, kept at the ELCA archives. Many ELCA church records have been digitized and are available via the subscription Ancestry site.