But by then, the original message, questioning the background of a presidential candidate who is of Indian and Jamaican descent, had already spread far and wide. For people like Mr. Alexander — an alt-right fringe figure who has also gone by the name Ali Akbar — the entire point of commenting was to go viral and counteract any progress made by a Democrat like Ms. Harris. She caught Joseph R. Biden Jr., the presumed front-runner, off guard on Thursday night by questioning his track record on race.

Because his tweet was elevated by valuable surrogates like the president’s son, Mr. Alexander has become part of a loose network of accounts weaponized by the Trump campaign as part of its effort to discredit candidates.

The facts are these: Ms. Harris, a first-term senator from California and a former prosecutor, is the biracial daughter of a Jamaican father and Indian mother. She has faced repeated questions about her race throughout her career, but has at times resisted being put into one category or another. In an interview with The Washington Post this year, Ms. Harris called herself “an American,” defying calls at various points for her to choose.

“My point was: I am who I am,” Ms. Harris said at the time. “I’m good with it. You might need to figure it out, but I’m fine with it.”

On Thursday evening, Ms. Harris was more forceful in relaying her experiences with racism. “Growing up, my sister and I had to deal with the neighbor who told us her parents couldn’t play with us because she — because we were black,” Ms. Harris said.