Thanks @CDBlue your reply
Since my question yesterday I restart my pc, and now everything work perfectly without any other changes...
So for people who would have the same problem as me, this is how I configured my Steam, Big Box and Dolphin:
In Steam Link:
Big Box Shortcut > Manage Controller > Gamepad Template
Controller Option > Uncheck the 2 options (Overlay & Desktop)
In Big Box:
Option > Controller > check both options (if they are not enabled natively)
In Dolphin:
Connect your controller(s) directly to the PC
Controller Settings > Port 1: choose "Standard Controller" > Configure button > Device: Choose Xinput/0/Gamepad > Map your buttons
In the same window, check the box "Always connected"
Do the same thing with all your gamepads and choose Xinput/2/Gamepad or Xinput/3/Gamepad or Xinput/4/Gamepad for each
Close Dolphin
/!\ For each controller, you must associate the Xinput / 1to4 / in the drop-down list. If you combine Xinput / 1 / gamepad with two different controllers,
for example, you can control 2 characters in a game with the same gamepad (the first one in the list). Not very practical!
And as @CDBlue said, check that Dolphin is not set to run as admin.
Now just to be sure, restart your computer ... And normally everything should work properly!