Since each blockchain is an island, asset transfer can currently only be achieved indirectly by means of exchange. This makes the term “cross-chain” somewhat of a misnomer. Assets do not “cross” chains.
For example, suppose you want to transfer BTC to Ethereum. BTC has a shadow token, tBTC, on Ethereum. An intermediary provides a pool of BTC on the Bitcoin blockchain and a pool of tBTC on the Ethereum blockchain. To “transfer” BTC to Ethereum, you “lock” the BTC into the intermediary’s pool on the Bitcoin blockchain and receive a corresponding amount of tBTC from the intermediary’s pool on the Ethereum blockchain. The intermediary uses a series of consensus to ensure that the amount of tBTC is the same as the amount of locked positions on BTC, but this still requires you to trust the correct behavior of the intermediary.