If a thing that is Invested under one Shard, you transfer it to another. Does the method of continuing empowerment of the Investiture change to the source of that other Shard?
Naturally, no. It may do so.
The only reason I ask is because Nightblood doesn't seem to behave different.
Yes, Nightblood does not require, but will instead accept gleefully anything you give it. But for instance, if you took a Soulstamp to another planet and somehow made it work, it wouldn't necessarily draw on the power of that Shard to work. Granted, it's really hard to make a Soulstamp work. Here's another example. You go on another planet. Hoid is using Allomancy on Roshar. That is not using the power of Honor or Cultivation. It is still drawing on the power of, in that case, Harmony.