Donald Trump’s adult children have spent a decade doing business with politically connected foreign firms—a role that will present potential conflicts of interest whether he wins on Nov. 8 or continues to pursue politics if he loses.

Those whom Ivanka Trump, Donald Trump Jr. and Eric Trump have worked with abroad include: the family of a developer in India who is a ruling-party politician, an Azerbaijani government minister’s son and a media company that became the Turkish president’s outlet of choice during the July 15 coup...