I have dyed several, British military surplus, nylon, water, ammunition and utility pouches with great success despite manufacturers claims that it couldn't be done with their (cotton) dyes. (they advised against using polyester dyes as being potentially dangerous - I didn't because the colour range was limited).
Most of the pouches were originally Brit desert dpm (sand/beige) but one was a Dutch (Brit) dpm. All were dyed to a dark brown.
My initial attempts were carried out as per the manufacturers instructions for hand dying cotton fabric - it didn't work!
I repeated the process but next time kept the dye solution hot, not quite boiling (over a burner outside) for about two hours and then left the pouches in the solution until it had gone cold - success! The pouches don't appear to have been adversely affected. I've been using them on and off for several years in all weathers and the colour has hardly changed.
Dying from a light to a dark colour will probably be most successful.
Be aware that the solution (dye/salt etc) might affect straps, buckles, stitching etc differently so if that could be an issue remove them, if you can.
BTW you've just reminded me that I must post some info about dying Ventile too