Elemental continuous abilities and weapon procs

There are many weapons that have a chance to proc some effect, usually damage, like all cache epics and Totemic Hell Lord. My question is this:
Do those procs work off non-initial hits of continuous elemental abilities, such as Crystallised Flame/Blaze, Crystallised Frost, Crystallised Storm, Blizzard and Inferno+passive? Or do those continuous portions not count as hits for the purpose of proccing weapon effects?

The only one I could easily find and test was Unstable Electron Core, which only works off fire and lightning abilities. And trying it with Crystallised Blaze/Crystallised Storm showed that only initial hits do the AoE proc, the continuous portion doesn’t seem to, at least according to combat log on dummies. Is it the same with other weapons that have such procs? I.E. will using Crystallised Blaze with Totemic Hell Lord only put 1 stack of DoT on target in the beginning instead of 10 by the end of it? Will only initial hit of Blizzard be able to proc Anima-Touched weapon effect?

The answer to your questions is yes. Only the very first hit counts as “hit”, the rest is treated as proc. That’s applicable, to my knowledge, to every ability in the game, including dots (like Eldritch Scourge). The reason is simple: an ability such as Crystallized Blaze still takes you 1 global cooldown to be cast, so there’s no reason its damage over time should also be able to proc anything besides the 1st hit. It gets 1 occurrence when it can proc something, like any ability in the game.

Thanks for your swift answer.
Then it means, if I only cast Blizzard and Crystallised Blaze from my offhand Ele Focus, investing in Anima-Touched is unwise. Totemic Hell Lord would probably be better for my goal of making a main/offhand capable elefocus, since it buffs Blizzard a bit less than Inverter, but has less maintenance reqs than Frozen Figurine. Lack of Blaze buffing (except for weak DoT) is something I’ll have to deal with.

You’ll only get 1 stack of the totemic fire dot with Blaze. The totemic has literally no use for a secondary.

It’s better then default one with it’s 32% bonus to Ice damage, seems to be less attention-intensive than Frozen Figurine (though I intend to try it out when building to red), and is not locked into off-hand like Inverter. So while not best at everything, or maybe even anything, it still has a use for both offhand and main hand.
I already try to keep all 9 weapons at level, so having 2 ele focuses, one for offhand, one for main hand is something I don’t want to attempt yet.

Maybe a Voltaic Shunt is what you are looking for ? It is a good secondary but I really don’t know how it would perform as primary. It gives like 0,3 CP dps if I remember well.

Shunt was meant more of an offhand, where you’d run yourself dry and let it naturally cooldown while you recover offhand energy.