"support them doing" is correct and usable in written English
You can use it when expressing support for someone's action. For example: "I support them doing whatever makes them happy."
The terrorists and the states that support them don't have large armies or precision weapons.
But too often they find that the services there to support them don't do the same.
Many adults who might otherwise be inclined to support them do not bother to vote, possibly because they do not regard the parties as viable and think voting would be a waste of time.
But, let's face it, the gun issue has its own unique dynamic, which is that the people who oppose gun limits vote on this issue while the people who support them do not.
What it is about lifting tens of millions of people out of poverty that Republican leaders and the fat-cat contributors who support them don't like?
It insists that the Palestinians, and the Arabs and Muslims throughout the world who support them, don't really care about the occupation.
However, those on Wall Street and government officials who support them do not want Warren to head the bureau, as they know that she would boldly stand up for consumers and crack down on the underhanded practices of big banks.
Gun violence is a public health and safety threat in the U.S., yet gun manufacturers, the NRA, and the political conservatives who support them, do everything possible to cover up the facts.
Support them, don't suppress: If you just get angry and try to control them then it's really not the correct way to handle the situation.
But I don't know this one.' I need to know who supports them, do they have an agenda, do they make a product that is related?" So perhaps the answer is not just more information.
