Asset management in a Chain Abstracted future is the same way the average person manages their bank accounts. 

1 account for spending, anywhere on anything. 
(This is a CAB account) 

1 account for specific spendings such as bills/rents etc 
(This is for assets on one chain for one purpose) 

1 account for long-term savings, slowly topping up. 
(This if for assets in the cold-wallet that don't move) 

TLDR is Chain Abstraction is giving more flexibility to the user as opposed to forcing them into rigid onchain asset management.