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Can't move portals.
Of course you can. They're on moon rock. There are a few puzzles with moving platforms with portals.
You can't place a portal on anything the game physics would allow to go through another portal.
Yeah but that's by game design, not internal world logic.
Theoretically you can if they're moving in a constant direction at a constant speed a la an inertial frame.
That doesn't work because gravity makes the game's reference frame non-inertial. One of the big takeaways of general relativity is that idea a reference frame that's inertial except for gravity is meaningless. Even ignoring relativity, everything is subjected to centripetal acceleration due to the Earth's rotation (and the story canonically takes place on Earth).
I think the real answer is probably the least satisfying: the game's physics just don't correspond to real physics. Most portals appear to exist in a privileged reference frame that can be said to be motionless, but even that isn't the real rule; the real rule is that portals can exist where the level designers want to allow them to exist. They try to make it feel like there's a certain logic behind it, but they'll bend the rules as necessary to make a cool puzzle work, and they keep the everything consistent within a single puzzle, but some subtleties of how portals appear to work are subject to change between puzzles.
As far as I'm aware gravity doesn't directly act on portals so I don't think they would experience acceleration from Gravity themselves. Though I was thinking about it more in terms of general relativity rather than Newtonian gravity.
Except that one time in Portal 2.
When you cut the tubes to the big gas chambery thing.