With movableTileEntities rule enabled and Lithium installed, hoppers will continue to transfer items to double chests after they have been moved away (piston, etc).

Tested in Minecraft 1.19.4

Carpet Mod - 1.4.101+v230219
Fabric API - 0.78.0+1.19.4
Lithium - 0.11.1
Mod Menu - 6.1.0-rc.4
No Telemetry - 1.7.0
Sodium - 0.4.10+build.24
Starlight - 1.1.1+fabric.ae22326