I have still never managed to catch my girls in their cage.
I have two options for moving them - one is to sort of chase/herd them into their long fleece tunnel, then I grab both ends and lift them out inside it. The other is to try and chase/herd them into a big snuggle sack and then lift them out in that.
They get a bit nervous and sometimes you have to lift out some of the other things they can hide in, but definitely rather try and herd them into a removable object and just transport them out in that as ultimately it is less terrifying for them and much easier for you.
ETA: by "chase" I mean slowly moving my hand up behind them. As soon as it gets near they dart off. This way I guide them into the general area of the tunnel or snuggle sack. Nia has realised now it means I'm going to take her out so she usually runs past it and does everything possible not to go in it at first, but normally within a minute or two she goes inside. It does often mean upturning all their other hidies first though.