The first classmate of mine to emigrate was Slavik Feierstein: in 1986, right after Chernobyl, he and his family left for the States. Many more - and not just my classmates - followed several years later.
Masha Gessen's family left the Soviet Union in 1981, after some three years of redtape and other obstacles. The airport farewell scene is one of the most moving in her book:
[...] I remembered our relatives, eyes red from sleeplessness and crying, lined up in two crowded rows against the chrome barrier that marked the border zone at Moscow's Sheremetyevo-Two Airport, their faces receding and approaching again. They watched two customs officers go through our belongings, packed into our 'emigrant's suitcases' - cheap cardboard affairs designed to withstand exactly one journey, one way. I ran back to hand one of my aunts my drawing box, barred by customs. I ran back again to take a drag from another aunt's cigarette. I was fourteen, but this was the most difficult day of our lives. We finally stepped through customs, sideways and backward so as to keep looking at their faces, receding now. They waved. Two-thirds of the way to passport control, just before their features became indistinguishable, my mother drew my head close and whispered, 'Look at them. You will see all of them again, except great-grandmother Batsheva. Look at her.' What she said was absurd: everyone knew there would be no return and we would never see any of them. [...]
Masha's parents and brother came back for their first visit in early 1988 - just as many people I knew were preparing for or going through their departures, an ordeal that must've been very happy and very heartbreaking at once.
In a way, the 1970s 'immigration wave' converged with the 1980's 'wave' somewhere in the air during perestroika, moving in opposite directions.