Do fruit effects transfer during inheritance?

The practical upshot is that fruit is an investment in your account, not in one creature. Feed a Kappa fruit to an Aniimo now and the effect is not stranded there when you later find something better in the same role. Pass it down through inheritance and the effect goes with it. Keeper fruit behaves the same way, and you do not have to do anything to make it happen. There is no separate transfer step, no confirmation, no fee attached to the fruit itself.

Capability Fruit upgrades ride along too. That matters more than it sounds, because Capability Fruit is also what raises an Aniimo's rarity tier. An Elite can be elevated to Perfect with a Capability Fruit that grants plus one Initial Potential in its recommended stat. So the work you put into a mid-tier Aniimo is not wasted when you replace it.

The upshot for a new player: stop hoarding fruit for the "right" Aniimo. Use it on what you are actually playing.

What the fruit costs before it ever transfers

Capability Fruit sells for 1,000 Glimmer in the shop, but you are not required to buy it. It drops through regular play, and you can exchange 20 Potential Seeds for one. If you are short on Glimmer early, the seed route is the one to watch, because seeds accumulate whether or not you are thinking about them.

Since the upgrade carries forward regardless, buying at full price to rush a tier upgrade on a current favourite is a reasonable use of Glimmer rather than a waste.

Not settled: Kappa, Keeper and Capability Fruit are the ones confirmed to carry over. Whether every other fruit type transfers on inheritance is not settled.