The idea of cross-chain activity is covered in full in the entry Cross-Chain Compatibility, which explains how different blockchains are made to work together and the approaches used. This page gives a short definition and explains how the word is used.
"Cross-chain" describes any transaction, application or protocol that operates across two or more separate blockchains. Because each blockchain keeps its own ledger and cannot natively see the others, moving value or information between them requires extra machinery: a blockchain bridge that locks tokens on one chain and issues representations on another, messaging protocols that relay data, or atomic swaps that exchange assets directly. Cross-chain tools let users move stablecoins and other assets to cheaper or faster networks and let applications span several chains. They have also been a major source of losses, since several of the largest crypto hacks have targeted bridges. More detail appears under cross-chain communication.
"Cross-chain" is the adjective, used in terms such as cross-chain bridge or cross-chain swap, while "cross-chain compatibility" and blockchain interoperability describe the property of networks being able to work together.