Following his first season with Tigers, Kayser won his first cap for France but was on the losing side in Australia, while Moreno celebrated his recall by Italy with a last-gasp win in Argentina.
Australia beat France 34-13 in their first Test in Sydney, with Kayser entering the action as a 50th-minute replacement for Sale Sharks hooker Sebastien Bruno.
Italy, meanwhile, scored a last-minute try to win 13-12 in Cordoba, with Moreno used as a 26th-minute replacement for Ignacio Rouyet.
The Azzurri has trailed 12-3 at half-time but a strong fightback was capped with the late score from replacement Leonardo Ghiraldini.
Moreno, who returns to Italy with Calvisano next season, collected his fifth senior cap for Italy after previously winning three caps for Argentina.