Note: I'm using examples for Germany in this answer because it is the country I'm most familiar on the continent, but the basic principles are the same for any country.
I'm going to start with the advice given by John P. Colletta, PhD., in his book They Came in Ships: A Guide to Finding Your Immigrant Ancestor's Arrival Record. Colletta suggests that you discover:
- Your ancestor's full, original name (much easier for those of you starting in Europe than it is for US researchers)
- Their age at arrival
- Their date of arrival
Since Colletta is preparing to help his readers search for passenger arrival lists, he is picking out the bits of information which is likely to appear in the search results for those arrival lists. We need to widen this to include any identifying features that will help you distinguish your research subjects when they are in a group of search results with other people who have the same name and are close in age.
So I would start by building a set of criteria that would help you do your searches. My list would include:
- Their name
- Their place of origin
- Any known occupations
- Their religious affiliations
- Their "last seen" data - their last known place and time before they 'disappear' from the records in Europe.
- Information about their family and FAN Club (friends, associates, neighbors).
If you haven't already, I would sort people into groups according to their place of origin, and the possible time frames for their emigration. The record groups you'll want to search in will differ, depending on the time period, and search strategies that work in 20th-century records may not be feasible for the earlier periods. All the research guides that are based for US audiences working in the other direction may help you, too. For example, the US National Archives' article Clues in Census Records, 1850-1930 summarizes what information can be found in which census years. Articles or classes that talk about the different waves of emigration from the country can provide useful clues for researchers on both sides of the emigration/immigration dilemma.
Next I would do a search in Google Books, Hathi Trust, the Internet Archive, Google Scholar, PERSI (the Periodical Source Index), and via Google or other search engines, to look for academic works, genealogy periodicals, histories, blog posts, or whatever else I could find, to find out background information and/or worked case studies showing where and how people from the country of origin came to the USA. Study the research methods of those researchers and see what sources they used, and how they used them -- read the works cited in their bibliographies to get ideas. In some cases, you can get spectacular results, such as the time when I found a book that was a revised edition of someone's doctoral dissertation about the German immigrant community in my husband's parent's hometown. From that and other writings, I learned how Germans in particular had been recruited for their skills as the US was becoming more industrialized. You may be able to find your missing person by following the trail of other people from the same community, since people often migrate in groups or practice chain migration.
Which record groups to focus on depends on the time period, but there are some things to consider, such as where are you most likely to find them. I'll use Germany as an example, looking up Werdau (Werdau, Zwickau, Zwickau, Sachsen) in the online Meyers Gazetteer. The website's creators have broken down in the detail section the different jurisdictions where the researcher might find records about people from Werdau, and what kind of records might be available at each level. If you aren't already familiar with United States jurisdictions, you might want to make a table or checklist like the detail section in the online MeyersGaz to sort out the what you want to focus on, because records can be found at all jurisdictional levels. Doing a place search in the FamilySearch Catalog, starting with United States, allows you to choose places contained within, working your way down to smaller and smaller levels. The online Atlas of Historical County Boundaries at the Newberry Libary, in the publications section, has information about when counties were created, and has shapefiles which you can download; the interactive map is down, but a similar county boundary map can be found at https://www.mapofus.org/.
If you are starting from zero and have no idea where your person may be in the USA, you may want to start with doing surname searches in the large Federal datasets that cover the entire country, such as the US Federal Census, and the passenger lists at the US National Archives' Access to Archival Databases (AAD), just to get an idea of where your most common surnames are found the records. Don't restrict yourself only to sites that are intended for use by genealogists -- academic projects such as Immigrant Entrepreneurship: German-American Business Biographies provide valuable context.
In some cases, like newspaper research, you may be able to find your missing people by searching for the place of origin, and sometimes you might find your missing person by searching for the names of the people who stayed behind in Germany. The Library of Congress has a portal for their collection at German American Newspapers and Periodicals; their project Chronicling America has featured articles such as Chronicling America’s Historic German Newspapers and the Growth of the American Ethnic Press. Ernest Thode's Historic German Newspapers Online, published in 2014, lists over 2,000 historic German-language newspapers online.
I'll add more ideas for a specific workflow as they come to me, but if you haven't already, start a research journal to assemble the information you already know, so you can look for patterns. In colonial times, sometimes entire congregations migrated together. That may not be the case so much for the 20th century, but being aware of the movement of clergymen or other community leaders might give you important clues.