That depends on how they make these children. As a Hero, you can produce Heroic children by conceiving (and possibly birthing) them in suitable magical environments. (Often that means: rites.) Choosing a suitably heroic partner certainly helps. Arim the Pauper underwent the Great Marriage with Sorana Tor, the result was the Twin dynasty. Sartar started his dynasty in his contests with the Feathered Horse Queen, founding a lineage of heroic individuals in Sartar (and probably among the Grazers, too).
Basically, to sire (or give birth) to (however lesser) demigods, you had best be divine upon the occasion. If those personal powers are variations of your cult's powers, make sure that the child is born into the cult. If you're really powerful (say: Red Emperor), make sure it is born into your cult (e.g. Arimtasus the Red Manimat).
> (Not meaning that they may
> pass along position and weapons, but rather the personal
> power they've gained).
The normal followers of a hero cult access (a watered-down variant) through cultic worship. I have often posed that descendants of a Hero might have another avenue - worship of a specific ancestor. While there is no real point for Heortling theists to do so for any old Joe Ancestor, it is a different proposal for Joe the Founder. Basically, if the ancestor has some identity remaining in the afterlife, it will be a heroic power.
Sartar the Hero is a special case as an ancestor manifest as a divine guardian.
> Which I guess is maybe just another way of approaching "If
> you gain power directly from the otherworld, does it become
> part of your nature (and so may pass on, like any other part of
> your nature), or is it always a 'borrowed' power, that you hold for
> only some period of time (even if that period is your life).
To have a power of your own, you need to make an impression on the Otherworld. The Otherworlds are mutable, so in order to make that impression last, you need to maintain that impression, e.g. by having a cultic followership linking to that part of the Otherworld from within Time. Being alive and using that power works, too. Being alive (within Time) and neglecting that power might result in the impression fading away, being usurped, etc..
> To put it in terms of hypothetical examples:
> - Able becomes a powerful priest and devotee, drawing great power
> from his god.
> - Baker becomes a powerful shaman, with a fetch and command over
> many powerful spirits.
> - Charlie gains control over his seventh breath while on a quest,
> and afterwards can blow out gale force winds.
> Able's magic all comes from his god. His child may also be become a
> devotee or priest, but has to get his own link to a god to get any
> magic, none of it inherits.
Unless the link is created with the siring and/or birth of that child. Which is why early mediaeval Germanic kings made a difference between legitimate sons and legitimate, purple-born sons, the latter having been sired when in office.
> Baker calls most of his magic from spirits, although his fetch is a
> part of himself. His child will not start off knowing any spirits,
unless Baker makes sure the spirits accompany the child already in the womb.
> but may also have the inherent ability to awaken his fetch--although
> that is still not inherited power so much as inherited potential.
Doing it right (or wrong), the child might even start as an embodied spirit. In that case, no need for it to become a shaman to access the spirit world.
> Charlie, I'm not so sure about. Would his child perhaps get a
> weaker version of this power? Would it depend on how Charlie
> gained the power?
To some extend, it will. We know that King Kocholang of Lankst comes from a family which can breathe fire. Someone must have brought this trait into the family...
Similar case: Blueface the Shaman, of the Elder Wilds.
Is it something that can be inserted into the reproductive process, or that the reproductive process can be inserted into?
All of these can opt for breeding for heroism, by choosing a magical wife (well, mother for the child). Able would likely father a child part his, part his god's; Baker's child might be not quite a human. Charlie might have to use his power to get into the reproductive act with a magical partner (who then would take care that that valuable trait may be inherited).
There is a minor chance that a casual layover for any of these with some tavern wench may produce a heroic child.
(Different for Harmast, whose layovers were of a ritual nature...)
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