Are higher rated capacitors interchangeable with lower rated ones?

Higher voltage ratings are probably ok, but it's not great generally to use higher capacitance values unless you know what its for. For the replacements, all you need to do is look for one that will fit on digikey that is 16V 470uF. Digikey will sell you most parts in single quantity. Try this list:

https://www.digikey.com/short/z9q7qt

It depends on the application, if the application is a power filter capacitor (used on Vcc of a component) then bigger capacitors have the same function. What is happening is we are forming an RLC filter. The LC components are formed by wires or traces. To prevent power dips sudden changes in Vcc we place a power capacitor to source power when the voltage drops (because the load changes), the value of C doesn't change the filter pole that much and bigger is usually better (but not if the ESR of the cap changes when we move to a bigger value).

When replacing power filter caps the voltage needs to be greater than that of the power rail, if you don't know what that is, then just make sure it's bigger than the cap. Make sure the ESR is equal to or smaller, and the value of the cap is equal to or bigger.

If the application is an audio filter or some other filter where the filter pole needs to be exact, then it's best to replace the cap with the exact parameters (ESR and component value) to get the same result.

So you could probably replace all caps on that board with the 470uF's (if you could figure out a way to make them fit) but not the other way around.

Higher voltage ratings are okay in and of themselves. But all things being equal, higher voltage caps have higher ESR and that might affect things if it's used in a switching circuit. You're probably good though unless the ESR is much higher or the circuit was borderline designed.