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Kaine also criticized Trump for calling his running mate, Democratic presidential nominee Hillary Clinton, a bigot.
"My reaction is we just ought to be extending our sympathy to the family. That's the only reaction that is appropriate right now and maybe a sadness about this gun violence issue, which we know it's complicated but that is, you see something like this and it's just, we should redouble our efforts to really adopt and promote smart strategies on that. But the sympathy issue is the one that ought to be our strong first reaction," Kaine said of Trump's tweet.
Dwyane Wade's cousin was just shot and killed walking her baby in Chicago. Just what I have been saying. African-Americans will VOTE TRUMP!— Donald J. Trump (@realDonaldTrump) August 27, 2016
Wade's cousin, Nykea Aldridge, was killed Friday in Chicago's Parkway Gardens neighborhood when two men exchanged gunfire nearby, according to police.
Kaine made the remarks about Trump's tweet following a tour of the small business, Design South Florida, in Miami Lakes, Florida.
"The tweet isn't important. What's important is this horrible crime, you know a woman and her child on her way to a store getting shot. I mean it's really, really tragic and of course you need good leadership to focus on these issues," Kaine said.
ABC News asked Kaine about remarks he made Friday in Tallahassee connecting Trump to "Ku Klux Klan values."
"He has supporters like David Duke connected with the Ku Klux Klan who are going around and saying Donald Trump is their candidate because Donald Trump is pushing their values. Ku Klux Klan values, David Duke values, Donald Trump values are not American values," Kaine said Friday at Florida A&M University.
Today, Kaine said he was not saying Trump had "Ku Klux Klan values."
"What I said yesterday was he's got guys connected with the Ku Klux Klan who are out, they are claiming him. And his record of, you know, sometimes he doesn't disabuse that and sometimes he seems to want to take advantage of that, and that I find that very troubling," Kaine said.
When asked earlier this year about what he thought about white supremacists, Trump told CBS News, "I don't like any group of hate. Hate groups are not for me.”
When reporters asked Kaine what was behind his intensifying attacks against Trump, he said that Trump's calling Clinton a "bigot" really bothered him.
"I mean I was pretty stunned by that," Kaine said.
He contrasted Clinton's career battling "for families and kids to get a fair shot in tough places" with Trump's remark about her.
"So that he is just going to casually say you're a bigot. Well then, OK. Well, it's important to call out what he's been saying, who he has been drawing support from. It was really important to do that," Kaine said.
The vice presidential nominee also criticized Trump's controversial overtures to the African American community over the past few days. Kaine said that Trump's past helping facilitate the "birther" controversy about President Obama makes it impossible for him to be serious about reaching out to the African American community.
"I don't see it as that serious because if you have been out pushing, promoting the notion that President Obama wasn't born in this country then you can say OK, well now I want to do outreach. I just don't see it as that serious," Kaine said.