You can use Connext to send transactions of value or calldata across chains and/or rollups. Unlike most other interoperability systems, Connext enables this without introducing any new trust assumptions or external validators.
Note that the current version of Connext cannot be used pass arbitrary event data between chains (i.e. prove to chain B that something occurred on chain A) as this cannot be done in a trust-minimized way in the evm except by incurring the 1-week exit window of existing rollup bridges.
This means that cases like migrating a token from chain A to chain B are not supported - you can instead only swap/transact over liquidity that already exists on the chain.