Here's Oxford on switch:
1 Change the position, direction, or focus of.
‘the company switched the boats to other routes’
1.1 Adopt (something different) in place of something else; change.
‘she's managed to switch careers’
[no object] ‘she worked as a librarian and then switched to journalism’
Switch and swap are closest, and they are synonymous with the set phrase "change out," though not interchangeable. It is hard to define these words without actually using them so I'm just going to describe contexts. You "switch trains", meaning you go from one train to the other. You can also "switch the channel" or "switch the settings on the computer.". You also "switch out/on the lights" (turn off/on the lights). It perhaps has a lot to do with things that involve a "switch" as a noun (something that makes a change, like a light switch).
Here's Oxford on swap
1 Take part in an exchange of.
‘we swapped phone numbers’
‘I'd swap places with you any day’
[no object] ‘I was wondering if you'd like to swap with me’
1.1 Give (one thing) and receive something else in exchange.
‘swap one of your sandwiches for a cheese and pickle?’
1.2 Substitute (one thing) for another.
‘I swapped my busy life in London for a peaceful village retreat’
Swap specifically refers to an exchange. It is more specific than any of them.
Change is probably the most broad word of the three and the least like the other two. I would rather not copy the definition from Oxford because it is broad. I think it generally refers to a physical state of change (he changed into a vampire) but it can overlap with the other two (he changed the setting on the computer). Honestly, I imagine you can use change in most of the above situations, except in the case of a binary "switch". You can't "change the lights," except when replacing the bulb. If you're changing the light colors, it also works.