The first crew transfer between space vehicles was from Soyuz 5 to Soyuz 4. It was done outside the spacecraft by Aleksei Yeliseyev and Yevgeny Khrunov.
The first U.S. crew transfer was during Apollo 9, between the command module and the lunar module. It was done both internally through the docking tunnel, and outside the spacecraft with a spacewalk by Rusty Schweickart.
Docking tunnels are now the standard method to transfer crew between vehicles.
What was the first (pair) of Soviet spacecraft to use a docking tunnel to transfer crew?