A Soviet "underground railroad"

No this is not an idea about the Moscow Metro. I've had this idea rattling in my head for a bit. What if there was a concerted effort in the west by an organization or otherwise to smuggle Soviet citizens, that obviously wished to leave, into America.

Please note I don't mean spys. I mean Soviets that actually wanted to escape persecution in the USSR.

Could this be done? How would it be done? Is it plausible or will it require a visitation from everyone least favorite space dwelling creatures of the Night?
 
I'm afraid that a concerted effort, meaning agents & middlemen in the USSR to organise things and contact the wannabe-refugees, will run into trouble with the KGB's 2nd*, 7th** & Border Guards Directorates. The problem is that for such an undergrounds to be effective, it has to be known to potential refugees or those likely to help them, this in turn guarantees that the KGB knows of it's existence. All they have to do then is to send the occasional fake refugee and interrogate caught refugees & agents to find the others. Rinse and repeat to catch different cells.

* Counter Intelligence
** surveillance of Soviet citizens & foreigners
 
It didn't happen in East Germany (post-Wall) which faced a far worse problem with emigration, so why would it happen in the Soviet Union itself?
 
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