WASHINGTON — Democrats, transitioning into the House majority, have quietly sent dozens of letters in recent weeks seeking documents and testimony from President Trump’s businesses, his campaign and his administration, setting the table for investigations that could reach the center of his presidency.

Clear targets have emerged in the process, and some others appear to have fallen away, at least for now. Family separation and detention policies at the border have jumped to the forefront. So has the acting attorney general’s oversight of the special counsel, Robert S. Mueller III. But Democrats, after slamming House Republicans for their inadequate inquiry, do not plan to reopen a full-scale Russian interference investigation. They have also chosen to hold off on an immediate request for Mr. Trump’s tax returns.

For eager liberals coming off two years of Republican oversight paralysis, the next few weeks may feel something like a game of hurry up and wait. Arranging witnesses and wrangling sensitive government documents take time, and most House committees have yet to be populated with lawmakers, not to mention much of the legion of lawyers who will do a lot of the work of investigations. The Intelligence Committee did not technically have a chairman until last week.

“Those people who are expecting some kind of Hollywood movie here are going to be disappointed because it is going be very orderly,” said Representative Jim Himes, Democrat of Connecticut and a senior member of the Intelligence Committee.