Lubimov has performed with such orchestras as the St. Petersburg Philharmonic, Royal Philharmonic Orchestra of London, Russian National Orchestra, Orchestre Philharmonique de Radio France, Toronto Symphony Orchestra, Deutsches Symphonie-Orchester Berlin, and Tokyo Symphony Orchestra, as well as the orchestras of Helsinki, Israel, Los Angeles, and Munich. He has worked with such conductors as Vladimir Ashkenazy, Neeme Järvi, David Oistrakh, Esa-Pekka Salonen, Marek Janowski, Christopher Hogwood, Sir Roger Norrington, Frans Bruggen, David Robertson, Andrey Boreyko, Ivan Fischer, Kent Nagano, and Yan Pascal Tortelier. In recent seasons he has performed with the City of Birmingham Symphony Orchestra, Austrian Tonkunstler Orchester, Orchestra of the Age of Enlightenment, Munchner Philharmoniker, SWR Stuttgart, Deutsches Symphonie Orchester Berlin, and Danish National Symphony Orchestra. Lubimov has recorded for many labels, such as Melodya, Erato, BIS, and Sony, which have released his interpretations of the complete sonatas of Mozart, as well as works by Schubert, Chopin, and Beethoven, and 20th century composers, including Arvo Pärt.

Since Merce Cunningham’s death in July 2009, Swinston has been the Director of Choreography, overseeing the Merce Cunningham Dance Company, the Cunningham Repertory Understudy Group and its work with the Cunningham Educational Outreach Program. Since 1998, Swinston has assisted in various Cunningham archival reconstructions including Suite for Five (1956–58), Summerspace (1958), How to Pass, Kick, Fall and Run (1965), RainForest (1968), CRWDSPCR (1993), and Ocean (1994), and the recent revivals of Squaregame (1976), Duets (1980), and Roaratorio (1983) for the Legacy Tour. He has assisted in the staging of Cunningham works on other companies, including Boston Ballet, White Oak Dance Project, Rambert Dance Company, and New York City Ballet. In 2009 Swinston was named a Trustee for the Merce Cunningham Trust.