Are there any Canon rules for how gravity affects characters?

Such as movement, strength, speed, carrying capacity, etc.?

Not in the original game. The original as you probably know is more about weight rather than mass or gravity.

If we were going to model it then I would give gravity an intensity in line with any other effect acting on a character or object. That way it also cuts down on the math and numbers and keeps it in the same style as the rest of the game.
Multiples of Earth gravity and their effects would then be calculated on the basis of Weight by Rank - a gravity creating the effect of tripling weight (3 Earth gravities) would be able to be looked up in the Weight chart and presto there's your Gravity intensity.

Zero G = Shift Zero / Zero

Average = 1 Earth gravity

Excellent = 2 Earth gravities

and so on.

I am using 1 adult = 1 Earth gravity so when the Weight in the Rank table is at 3 adults - 3 gravities. After a few more Ranks as with all other ratings in a FASERIP system it goes geometric / logarithmic / arbitrary but for a superhero game I still think that's totally fine. Trying to add a detailed mathematical system into FASERIP raises soooo many questions. A bit like the Ranged vs Area in the advanced rules, measuring tapes and other wargamey type stuff I just don't think is a good fit for FASERIP.

It's nice as well to be able to just say "After entering the upper atmosphere of Jupiter, the character are immediately subjected to Remarkable intensity Gravity."
It fits in with Gravity Control type powers better that way too.

I agree regarding matching gravity intensities to the rank system. Not a bad suggestion for approximating, thanks.

I think Time Traveller has a good recommendation. I would also say that the effect of the gravity ranks would be additive or subtractive according to a gravity control power. So a Typical rank gravity control would double the gravity on you on Earth. Or make you functionally weightless.