When democracy is at its best, elections are the civilized version of warfare: clashes of wit, personality and ideas that seek to resolve a nation’s biggest arguments. But sometimes an election is just a dirty fight.

When Donald J. Trump arrived in Albuquerque last week, stepping from his gold-plated jet into an aircraft hangar that was thronged with baying supporters, he held his arms aloft like a champion boxer.

His speech opened with dutiful attacks on Obamacare, the Iran nuclear deal and immigration. But it was when he uttered a single word — “Hillary” — that the crowd truly erupted.

“Lock her up!” The roar filled the hangar. Mr. Trump paused his speech, then smiled.

Two days later and 1,500 miles away, Hillary Clinton punched back. Slamming her rival’s “dark and divisive vision,” she alternated between mocking Mr. Trump and painting him as a threat to America and global stability.