Oy, I read some legend saying that kitsune for thanking human when they help them leave behind them eternal FIRE in the local temple, so they can use FIRE.
So, let's go for the ability tour of the kitsune.
Doton, since you are so forcefull about it, is their prime ability, no problem with that.
Mokuton, Inari is one of the forest protector, so kitsune being her spawns protecte forest and use its power in battle.
Katon, it is said that kitsune, for thanking humans that hepl them, leave behind them eternal fire in the local temple, so fire IS one of the kitsune ability.
Suiton, no special ability with water, it's neutral.
Raiton, don't know for this one, but I think it's like suiton.
Futon, look at Suiton and Raiton.
Hyoton (ice if I cirrectly remember), absolutly no talent in that area, the whole prospect of ice kill life, wich Inari is one of the protector, making kitsune illaligned with that element and by extention with the Yuki Onaa (who are know to bring "eternal winter" in their vincity).
Steel (don't know the something-ton that name it), no information about this one, but being forest protector, I don't think that kitsune are good in that area.