Walz-Vance debate updates: VP candidates tangle on abortion, immigration and Jan. 6

Vance said a few times in this debate that Republicans need to "win back people's trust" on the issue of abortion, and the polls support that.

Abortion is important to many voters this cycle, especially women and Democrats. And they trust Harris more than Trump on the issue. In an ABC News/Washington Post/Ipsos poll from August, they gave Harris a 12 point lead on the issue over Trump.

Since the U.S. Supreme Court's decision overturning Roe v. Wade, more Americans have also expanded their view of abortion rights and don't want states to put limits on the procedure. KFF has found that nearly three-quarters of reproductive-age women oppose leaving abortion access up to the states, as Trump and Vance have said they support.

Democrats have been working in states across the country to paint Republican candidates as too extreme on the issue, and are hoping it drives voters to the polls as it has the past two years.

—538's Monica Potts

Fact Check: False

When running for Senate in 2022, Vance signaled support for a bill introduced by Republican Sen. Lindsey Graham that would ban abortion after 15 weeks nationwide. Vance called it "totally reasonable to say you cannot abort a baby, especially for elective reasons, after 15 weeks of gestation."

Also during his 2022 Senate race, Vance said he supported setting “some minimum national standard” in an interview with USA Today.

—Cheyenne Haslett

At times getting personal about home ownership, Walz defended Harris' plan to build three million new homes.

"I think we should be able to find some common ground," Walz said. "But we can't blame immigrants for the only reason that's not the case that's happening in many cities, the fact of the matter is, is that we don't have enough naturally affording affordable housing, but we can make sure that the government's there to help kick start it."

Vance continued to claim illegal immigration is responsible for higher prices.

"Tim just said something that I agree with. We don't want to blame immigrants for higher housing prices, but we do want to blame Kamala Harris for letting in millions of illegal aliens into this country, which does drive up cost," he said.

CBS News' moderators then asked Vance for evidence that illegal immigration is leading to higher home prices. He said there was a "Federal Reserve" study he said he would share after the debate but did not elaborate.

The candidates talked about their own children as the debate turned to gun violence and Vance was asked if parents being held responsible could curb mass shootings.

"I want to just sort of speak as a father of three beautiful little kids, and our oldest is now in second grade," he said. "Like a lot of parents, we send our kids to school with such hope and such joy and such pride at their little faces on the first day of school, and we know, unfortunately, that a lot of kids are going to experience this terrible epidemic of gun violence."

Vance said he thinks part of the answer is to increase security in schools.

Walz said his 17-year-old son, Gus, witnessed a shooting in a community center playing volleyball and spoke about how meeting with the parents of Sandy Hook victims impacted his views on banning assault weapons. He also mentioned how he and Harris are gun owners.

"We understand that the Second Amendment is there, but our first responsibility is to our kids -- to figure this out," he said.