In 1970 he founded the Moscow string quartet. He was its member until 1979. In 1981 Boris Kuschnir moved to Austria. In 1984 he became a professor at the Vienna Conservatory, in 1999 – a distinguished professor at the University of Music and Theater in Graz.
In 1984 Kuschnir founded the Vienna Schubert Trio, which was awarded numerous prestigious awards, among which there are the first prize of the Sergio Lorenzi International Competition in Trieste (Italy), the "Mozart Interpretation Prize" in Vienna and the prize of the Ernst von Siemens Foundation. In 1993 he founded the Vienna Brahms Trio, which successfully debuted at the Gidon Kremer Lockenhaus Festival . In 1996 the Trio won the first prize at the 9th International Competition of Chamber Ensembles in Illzach (France). The recording by the Vienna Brahms Trio of all R. Schumann's piano trios was released by the label "Naxos" in 1999.
In 2002 Boris Kuschnir was one of the founders of the Kopelman Quartet, with which he tours and gives concerts around the world and has released a number of recordings at "Nimbus Records" and "Wigmore Hall Live". In 2008 the President of the Republic of Austria awarded Boris Kushnir with the Grand Decoration of Honor in Silver for Services to the Republic of Austria. The Austrian National Bank has provided Boris Kuschnir with the violin by Antonio Stradivari (La Rouse Boughton, 1703) for long-term use in token of recognition of his contribution to the art of music.