Interoperability is one of the main challenges of blockchain technologies, which are generally designed as self-contained systems. Interoperability schemes for privacy-focused blockchains are particularly hard to design: they must integrate with the unique privacy features of the underlying blockchain so as to prove statements about specific transactions in protocols designed to obfuscate them. This has led to users being forced to weaken their privacy, e.g. by using centralised exchanges, to move assets from one chain to another. We present ZCLAIM, a framework for trustless cross-chain asset migration based on the Zcash privacy-protecting protocol. ZCLAIM integrates with an implementation of the Sapling version of Zcash on a smart-contract capable issuing chain in order to attain private cross-chain transfers. We show that a tokenised representation can be created via a set of collateralised intermediaries without relying on or revealing the total amount to any third party.