Background: My wife was raised in one of these groups.

Fundamentalists believe in gathering, so there aren't very many outside the United States (with the exception of Canada and Mexico). Any effect they have on the Church comes from being concentrated near headquarters, not from influencing everyday members around the world. That said, I can think of several ways that Fundamentalists have influenced church policy:

• The temple recommend question about affiliating or sympathizing with groups opposed to the Church was added to weed out Fundamentalists.
• Ditto the Strengthening Church Members Committee.
• Elder Poelman's 1984 talk about becoming less dependent on the Church as you advance in the gospel was heavily edited to preempt Fundamentalists from making use of it.
• The November 2015 policy toward the children of gay parents was modeled on longstanding policy toward the children of polygamists.

In general, I've found that a lot of things liberal Mormons think were made to target them were actually implemented with Fundamentalists in mind. While most members pretty much never think about polygamists, the GAs are very much aware of them (converts from the Church to Fundamentalism have to be interviewed by a member of the First Presidency).