Here are a couple examples of initially neutral or moderate organizations that became left wing bastions:

Academy of Motion Picture Arts and Sciences

Screen Actors Guild–American Federation of Television and Radio Artists

Writers Guild of America

Directors Guild of America

National Education Association

American Federation of Teachers

American Psychological Association

American Sociological Association

Modern Language Association

American Historical Association

Association of American Universities

Columbia University

Harvard University

Yale University

University of California, Berkeley

Oberlin College

Reed College

NPR

PBS

The New York Times

The Washington Post

The Guardian

Human Rights Campaign

GLAAD

Good luck finding examples of the opposite direction.