Someone correct me if I'm wrong, but most DAW's process mixing at a high internal bit-rate regardless of what your project or files are when you bring them into the app. You said you used Audition, but didn't say if you independently processed the files by themselves, or if you used multi-track mode.
At least with the newest Audition 3 (not sure about older versions but I bet this applies also), you can set what bit-rate to process multi-track mixing/mixdown at up to 32-bit. Select Edit > Preferences (or hit F4) and go to the the Multitrack tab.
Keep in mind that if you are manually upconverting your files to 32-bit, working with them at 32-bit then exporting, you will need to dither when converting the audio back to 24 or 16-bit.
Specifically addressing your question, I'm fairly sure I remember Bob Katz saying somewhere that you shouldn't need to upconvert your files, as the DAW processes everything at a high internal bitrate anyway. Unfortunately, you may be at the mercy of the DAW's internal dither when coming down from 32-bit back to 24 or 16-bit. I would recommend rendering your file at at least 24-bit after editing, and then running it through the best dither you have available.
In Auditon's case, if you have a VST dither you want to use (like iZotope Ozone's excellent MBIT+ dither), slap it on as the last VST effect on the master bus and export to your target bitrate.