Actually there's also the body weight, the wood age, the finish and countless different things that affect tone. We can, however, take a good guess how will something affect tone, this is what OP asked for. We ain't here to overcomplicate things to the OP, we're here to help and if in any way possible simplify his/her search for the tone he/she is looking for. I said what ash sounds in general, what he can expect in general. There's whole other thing that, to my ears, DiMarzios are much less picky (again, in general - then some other manufacturer's pu) as far as different woods go, specifically in same species. It's a fact that simply saying "can't help you until you try yourself" isn't helpful.
I am fully aware of all that, and I certainly did not want to overcomplicate things. I would agree that swamp ash tends to have more of a midscoop which usually accentuates low and high frequencies. But if the midscoop is in a certain part of the midrange, the guitar can sound very dark. I have such a guitar with a swamp ash body that has boomy bass, is dark in the midrange and has good but not overbearing highs. The pickups typically recommended for swamp ash (strong mids and lows) all failed miserably in this guitar and produced muddy bass even on the bridge pickup. The only combination that worked was Air Norton in the bridge and a Bluesbucker in the neck because they were bright enough to balance out the inherent darkness of that guitar.
I have never had a similar experience with an alder or mahogany bodied guitar. Neither wood usually does exhibit a midscoop as does ash.
As for the rest I feel you gave the correct advice to the OP.
Cheers Stephan
Yeah, some woods are more uniform in the same species (mahogany is more uniform as far as sound goes even cross species - ie khaya vs honduran), I haven't yet stumbled upon as finicky piece as you did (well, I did with Sen ash, but Sen isn't really a type of ash at all so that's besides the point), I do agree such pieces are a special case, I am sad you got one such on your guitar. In general though, that's a minority.
There's another thing that perhaps many will disagree with me on, but certain pickups do excellent job in masking the guitar's tone. In general, a large magnet ceramics do this the best (aside the obvious active pickups) to my experience, but also, for some reason, the Air versions of some higher output pickups (as far as DMZ goes) as well. I'm not saying Air Zone or Air Norton sound the same in mahogany alder or ash, but I am saying that, for some reason, these pickups tend to filter the negative aspects (overbearing highs of maple, muddy lows of specific heavy mahogany piece etc) of a finicky guitar wood much better, giving a predictive sound that is quite similar in many different woods. This might explain why you liked the AN on the bridge as much, since it remedied some of that muddiness/darkness of the wood without adding any unnecessary brightness.
Just my experience anyway.