The flash point was Thomas the Tank Engine.

Last September, the National Rifle Association’s famously combative spokeswoman, Dana Loesch, provoked widespread outrage when she took to the gun group’s streaming service to mock ethnic diversity on the popular children’s program “Thomas & Friends,” portraying the show’s talking trains in Ku Klux Klan hoods. Now, growing unease over the site’s inflammatory rhetoric, and whether it has strayed too far from the N.R.A.’s core gun-rights mission, has put its future in doubt.

The site, NRATV, is a central part of the organization’s messaging apparatus. Since its creation in 2016, it has adopted an increasingly apocalyptic, hard-right tone, warning of race wars, describing Barack Obama as a “fresh-faced flower-child president,” calling for a march on the Federal Bureau of Investigation and comparing journalists to rodents.

In recent weeks, in a rare airing of internal debate at the N.R.A., two prominent board members expressed concerns about NRATV to The New York Times. Their statements were released through the N.R.A. itself, amid what was described as an internal review of NRATV and its future.

“Since the founding of NRATV, some, including myself and other board members, have questioned the value of it,” Marion Hammer, the group’s most formidable lobbyist and a key adviser to its chief executive, Wayne LaPierre, said in a statement. “Wayne has told me and others that NRATV is being constantly evaluated — to make sure it works in the best interest of the organization and provides an appropriate return on investment.”