Smith joined Walmart in July 2011 as chief marketing officer in China. He will take up his new role in May.
Smith has 10 years of marketing and merchandising experience in the grocery business, working with Delhaize Group in North America and Belgium.
He has also held marketing and advertising roles at AT&T, sports media agency Resort Sports Network and J Walter Thompson.
Walmart said Bendel would take on a "broader project-based role for Walmart" after being influential in building its global marketing efforts as well as heading up the Asda marketing department.
Andy Clarke, Asda president and chief executive, said: "Because this is a time of strength for Asda, it now seems the perfect time for Rick to do more to help shape Walmart's worldwide marketing efforts and embark on the next exciting phase of his career."
Follow Matthew Chapman at @mattchapmanUK