In tournaments, she took 1st-3rd at Sukhumi Candidates Tournament (joint Milunka Lazarević and Tatiana Zatulovskaya) 1964, won at Beberjik 1967, won at Subotica (Candidates Tournament) 1967, 2nd at Belgrad 1968, tied for 1st-2nd (with Nikolau) at Sinaia 1969, tied for 2nd-3rd (with Vobralova, won by Ivánka) at Wijk aan Zee 1971, won at Belgrad 1971 (ahead Gaprindashvili), won at Moscow 1971, won at Vrnjačka Banja 1973, 3rd at Voronezh 1973 (behind Zatulovskaya and Saunina), won at Roosendaal Interzonal 1976 (joint Akhmilovskaya).
Kushnir was a three-time winner of the Women's Chess Olympiad: in 1969 and 1972 she won the tournament as a player in the Soviet team, both times showing the best result at the 2nd board,[3][4] and in 1976 she won it as a player in the Israeli team, showing the best result at the 1st board.[5]