I don't think any FB papers use incorporated developing agents, and I believe Ilford stopped using them in RC papers with MG III or MG IV RC for archival stability reasons.
I've never noticed a problem from running different papers through the same tray, and I find Neutol-WA to be a long lasting developer (1+7 dilution), but I have a theory. It might be that one paper is more sensitive to the weakening of the developer than the other. As you run paper through the developer, bromides build up as a byproduct of development, resulting in longer development times (one solution to this is factorial development, described in Adams' _The Print_). With some paper-developer combinations, you can add potassium bromide intentionally for warmer tones. So maybe with the developer you are using and the warmtone paper, you are getting more "warmth" (turning toward olive) as the bromide builds up in the developer, and maybe the cold tone paper just doesn't react strongly in this regard, so it seems more stable.