question on defecting from the soviet union

Was watching “Moscow On The Hudson” on HBO the other day, and I have a question about defections from the former Soviet Union.

In the film, the Russian Circus is visiting NYC, and one of the clowns wants to defect. He arouses a lot of suspicion, and his KGB handlers keep close tabs on him. While they are watching him, a character played by Robin Williams chooses to defect while shopping at Bloomie’s.

To make a long story short, the KGB try to talk him out of it but he claims political assylum and the cops basically say it’s NYC, and they can’t force him to get back on the Russians’ tour bus if he wants to defect.

Williams then goes outside as the bus drives away and waves goobye to his friend stuck on the bus, the one who wanted to defect originally.

So here’s my question: was it that easy for Russians in the US to defect? Forget about the consequences to their family left behind… if a Russian performer merely told the authorities in NYC that he wanted to defect, would that work?

And if it would, what was stopping the other defector-wannabe from merely stepping off the bus and saying, “Me too!”

Yes, I know it’s just a movie. Which is why I’m looking for the Straight Dope.

PS - yes this did take place during the Cold War years.