If the flood was recent and local, then when did humans become omnivores?

After the flood, in Genesis 9:2-3, God gives humans permission to begin eating animals. We are finding traces of human civilization going back tens of thousands of years for Native Americans, so they must have gotten to the Americas before the flood. If the flood was local and in the last 10,000 years, then when did the Native Americans and people on other continents switch to eating animal meat?