DES MOINES, Iowa—The political shadow of former President Obama will loom large this week when most of the Democratic presidential field gathers inside a 16,000-seat arena in this city’s downtown.

Mr. Obama’s performance at a similar event 12 years ago—including his well-received speech and his campaign’s showmanship—helped him move up in the polls and dent the perceived inevitability of then-Sen. Hillary Clinton’s 2008 White House bid.

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