The Republican debate on Tuesday evening was set for fireworks, but the first real explosions of the night came from an unexpected source.

A substantive debate on economics and tax policy descended into petty insults when Ohio Gov. John Kasich and Donald Trump squabbled over the issue of whether millions of undocumented immigrants in the United States could be deported.

Kasich, who interrupted the moderators multiple times throughout the debate to interject his views, rebutted the idea that a future President Trump could deport millions of immigrants after the construction of a border wall between the United States and Mexico.

“In 1986, Ronald Reagan basically said, the people who were here, if they were law-abiding, could stay,” Kasich said, citing the conservative icon.

Kasich then blasted Trump for making a “silly argument,”opened the door for the government to hold back pensions when people are convicted of national-security crimes also stemmed from an individual case. Alger Hiss, a former State Department official, was accused of spying for the Soviet Union in the 1940s. Mr. Hiss served a prison sentence for perjury related to an investigation of the spying allegations. Congress passed its law on revoking pensions, often referred to as the Hiss Act, in 1954.

Mr. Hiss, who died in 1996, denied the allegations throughout his life.

Write to Christopher Weaver at christopher.weaver@wsj.com and Dan Frosch at dan.frosch@wsj.com