"The kids came all around pretty quickly, but each of them had to process it in their own way," a source tells PEOPLE

After his secret struggle, reality TV father and former Olympian Bruce Jenner is "finally happy" and transitioning into a woman. Subscribe now to read the REAL story behind his life-changing decision in PEOPLE.

When Bruce Jenner was finally ready to tell his children that he is transgender, his revelation was met with less surprise than he may have anticipated.

“Bruce told them everything – where things stood, what his plans were for the future, and what everyone could expect,” a family insider tells PEOPLE of Jenner’s decision to transition and live his life as a female. “Everyone basically knew what was up, but it was still a shock to hear it outlined so explicitly.”

But it didn’t take long for Jenner’s six children – and his ex-wife Kris‘s kids Kourtney, Kim, Khloé and Rob Kardashian – to rally around the patriarch of reality-TV’s most famous family.

“They all came all around pretty quickly, but each of them had to process it in their own way,” says the insider. “All they want is for him to be happy.”

For Kim, the process included checking in on how her mother was responding to the news.

“Kim went to Kris and asked her how she felt. They had a very long conversation about it,” says the insider. “Once Kim was satisfied that Kris was emotionally okay, she was able to give Bruce her unconditional support.”

As for the men in the Kardashian women’s lives, the insider says Kourtney’s boyfriend Scott Disick “didn’t react very well” at first, but “is now on board” while Kim’s husband Kanye West seemed to embrace Jenner’s transition immediately.

“He is totally fine with it,” the insider says of West. “He believes in live and let live, and he took it all in stride. He and Kim are just trying to show Bruce love and acceptance – complete acceptance.”

For more on Bruce Jenner’s transition, pick up the latest issue of PEOPLE, on newsstands Friday

And for more information about transgender issues, visit the Human Rights Campaign, GLAAD and PFLAG.

Reporting by STEVE HELLING