One could use the Behringer as a submixer, sending the main L/R output to the Duet's 1/4" inputs.
Pro's: Ability to use as many inputs as the Behringer has.
Con's: Inability to edit the individual signals once they are submixed. You also can't use both of the Apogee's mic preamps and 1/4" inputs simultaneously. You
could use one preamp and one 1/4" for a mono submix at the same time, but you have to pick two inputs (vs. four).
This makes sense to me as I'm writing it, but it may not translate well.
I could see doing this for something like drums. Perhaps you come up with a mix of drum mic's you're happy with, and just record the stereo mix of drums. Same for ensemble work... say, four mic's on a string quartet tracked in stereo.
More importantly, however, is the quality of preamp. No offense to Behringer gear, but it's not in the same class as Apogee. Using a small Behri mixer in front of the Duet is like having a Mercedes and putting K-Mart tires on it. If you truly need more simultaneous inputs and want to preserve the audio quality you're getting now, you might look at upgrading to
the Apogee Ensemble.
For submixing a set of drums, the setup described would be a fine solution.