During a debate on Sunday, Senator Chuck Schumer of New York pronounced himself “appalled” — at least four times — that the director of the Federal Bureau of Investigation had disclosed that he was again looking into the matter of Hillary Clinton’s emails without yet knowing the significance of a newly discovered batch.

“He doesn’t even know what was in them, and yet he goes ahead and writes a letter,” said Mr. Schumer, a Democrat, adding that the director, James Comey, owed the nation a fuller explanation.

Mr. Schumer’s Republican challenger, Wendy E. Long, responded by arguing that Mr. Comey had a duty to evaluate any new evidence in the inquiry. And she added a knife twist, noting that the new emails had come to the F.B.I.’s attention amid an investigation into sexual messages sent by Anthony D. Weiner, who was, Ms. Long told Mr. Schumer with a smile, “your old protégé and friend.”

So it went at Union College in Schenectady, N.Y., where the two candidates for the Senate often appeared to be waging a proxy war for each party’s presidential nominee and their records as often as discussing their own differences in the debate, sponsored by NY1.