How to clean cage?

How do I clean the bird cage/change liners with my birds still inside? I got them very recently so they don't trust me yet and I can't get them out to clean. They won't leave on their own and I don't want to force them. This is the current cage I'm using, I have to unlock the bottom parts and pull the entire top off in order to get to the liners which disturbs the birds and I don't know what else to do.

I would greatly appreciate any help or advice, thank you.