Cross-signed hybrid certificates enable a migration path for systems with multiple components that cannot all be upgraded or replaced at the same time. This type of hybrid key exchange allows you to transition from traditional public-key cryptography to post-quantum cryptography in a more manageable way. Using hybrid certificates is akin to a house with two doors where each door has its own separate key.
If you install a new front door lock, only people with the new key can open that door. People with the old key can still enter the house, but only via the unchanged back door. Over time, you can swap out keys to users, giving them access through the new door lock. Eventually, once everyone's key is swapped out, you can safely change the back door lock with no loss of access for anyone.