In order to export your mix from Mixer to Bridge you first have to make sure that both applications are pointing to the same asset library on your machine.
You can do this by opening both Bridge and Mixer together.
In Bridge, go to Edit > Settings to open the Settings pop-up.
In Mixer, go to Edit > Preferences to open the Preferences pop-up.
The Local Library Path field in Bridge Settings, and the Local Library field in Mixer Preferences should be the same.
To send your mix to Bridge, in Mixer, go to File > Export to Library:
On the pop-up that opens, enter the required information and click Export:
Go to Bridge, and you should be able to see your exported mix in the Mixer tab.
Note: if you don't see it, try refreshing Bridge (CTRL/CMD + R)