Kalunga,

The Japanese Browning Low Wall was chambered for the .243 Win and the .260 Rem., so I would assume the action is strong enough for a 7x57, which is operating at lower pressure.
Pressures of 9.3x62 and 9.3x74R are even lower. But I agree, sticky actions and extraction may be an issue.
A good gunsmith should know what can be done.

Fuhrmann