Kaut first represented the Czech Republic at the 2016 World U-17 Hockey Challenge, in a 7th place finish. Kaut had a goal and three assists in five games as a 16-year old and a member of his country's first gold-medal at the 2016 Ivan Hlinka Memorial Tournament. Kaut emerged as a standout at the 2018 World Junior Championships in Buffalo, New York, where he ranked fourth on the team with 7 points in 7 games en route to a fourth-place finish at the tournament. His five assists were tied for third on the team and second among the Czech forwards, while playing on the top scoring line alongside Filip Zadina and Martin Nečas. Kaut returned for the 2019 World Junior Championships in Vancouver, British Columbia, after he was loaned during his first professional North American season within the Colorado Avalanche organization. In the opening game of the Tournament, Kaut scored and added the primary assist in a 2-1 overtime victory against Switzerland, to be named player of the game, on December 26, 2018. Helping the Czech's advance to the knockout rounds, Kaut scored the lone goal in a 3-1 elimination defeat to the United States on January 2, 2019. He was selected among the top 3 players for the Czech team in a 7th place finish.