David Welch, a former GOP National Committee research director and campaign adviser to John McCain, is among a faction of relatively moderate Republicans who wish the party was keeping pace with the Democrats on same-sex marriage. But he doubts that the emphasis on gay rights at the Democratic convention will provide much of a boost for Obama.

‘‘They’re playing into the narrative that Barack Obama can’t run on the economy, and this election has to be about the politics of division,’’ he said.

Kenneth Sherrill, professor emeritus of political science at Hunter College in New York City, said in an online column Tuesday that the Democrats should not take gay and lesbian support for granted. He urged the GOP to compete for their support.

‘‘The Republicans must learn to appeal to LGBT voters on the basis of promises that Republicans in office will pursue policies that will be good for the LGBT people,’’ Sherrill wrote for the Bilerico Project, an online aggregator of gay-rights blogs.

‘‘And, frankly, this will be good for the LGBT people,’’ he wrote. ‘‘If the Republicans were to act rationally and make a serious effort to get LGBT votes, Democrats would have to be better. Our community can only benefit when both parties fight for our votes.’’

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