Falvey was the expedition leader of the first Irish team to reach the summit of Cho Oyu in China and Nepal without oxygen, on 20 May 1998. He reached the summit of Ama Dablam in Nepal on 3 November 1999. In 2003, Falvey was the expedition leader of the first Irish team to reach the summit of Mount Everest via the South-Southeast Ridge, with team members Ger McDonnell and Mick Murphy reaching the summit. On 18 May 2004, Falvey reached the summit of Everest via the South-Southeast Ridge, and led the expedition that saw the first Irish woman, Dr Clare O'Leary, reach the top of Everest.