GREENVILLE, S.C. — Three prominent female Democrats all but openly began running for president this week, taking their most active steps yet to challenge President Trump and claim leadership of a movement of moderate and liberal women that has come to define their party during the 2018 elections.

Senator Kamala Harris of California on Friday took the stage in a church in the early primary state of South Carolina, as an audience chanted: “Madam president!” A day earlier, Senator Kirsten Gillibrand of New York visited similarly crucial New Hampshire, calling the November election a pivotal moment for women. And Senator Elizabeth Warren of Massachusetts left little doubt about her intentions when she released a genetic test indicating she has Native American ancestry — a move to blunt Mr. Trump’s taunts alleging she had mischaracterized her heritage.

These women are beginning to offer themselves as potential presidents at a time when stark divides around gender are shaping the midterm campaigns: A record number of women are running for Congress, mainly on the Democratic side, and polls show women favoring Democrats by a huge margin. Yet Mr. Trump has begun sharply assailing the #MeToo movement and making increasingly explicit appeals to male identity.

Ms. Gillibrand, who promoted a paid family leave proposal beside Molly Kelly, New Hampshire Democrats’ nominee for governor, in a Concord candy shop on Thursday, predicted multiple women would run against Mr. Trump in 2020. She said she had made no decisions about her future, but cast the political moment as one of women mobilizing against a “credibly accused sexual harasser and sexual assaulter” — Mr. Trump.