What you refer to for tx verification are “chain relays” (name first mentioned in V. Buterin’s report on chain interoperability ). The principle is the same as for SPV/light clients. In theory, every chain can implement an SPV client for another chain, independent of the underlying consensus mechanism - in practice this is way more complicated and requires a case-by-case feasibility analysis.
First, some details on PoW chain relays.
There exist PoW relay implementations for:
In our recent paper (XCLAIM) we provide a more detailed discussion of the security, functional requirements and practical challenges of chain relays for PoW blockchains: https://eprint.iacr.org/2018/643.pdf (Sec. V-B, Sec. VII-A,B,D and Appendix D). Another relevant paper is “PoW sidechains”
PoS Chain Relays
You implement similar constructions for PoS blockchains. The verification is likely similar to the case of PoW blockchains, however the functional requirements of a such chain relay are slightly different. Specifically, instead of verifying the PoW target against the difficulty (which implies that the relay must implement the difficulty adjustment mechanism), the relay instead must verify the signatures of the PoS committee. From this follows, that the relay must know of the staking mechanism and the distribution at every round/epoch (incl. identities the elected committee).
I have not yet seen an implementation of a such relay, however “PoS Sidechains” by Gazi, Kiayias and Zindros provide some very interesting work.
Hope this helps.
Note: there is a significant difference between “real world” data feeds and cross-chain verification is that we can provide cryptographic proofs for events occurring in blockchains, but (likely) not for events occurring in the “real world” (e.g. a horse race: you must trust the provider of the results).
 Buterin, Vitalik. “Chain interoperability.” R3 Research Paper (2016).
 Zamyatin, A., Harz, D., Lind, J., Panayiotou, P., Gervais, A., & Knottenbelt, W. J. “XCLAIM: Trustless, Interoperable Cryptocurrency-Backed Assets”. To appear at IEEE S&P 2019.
 Kiayias, Aggelos, and Dionysis Zindros. “Proof-of-Work Sidechains”. 3rd Workshop on Trusted Smart Contracts at FC 2019.
 Kiayias, Aggelos, and Dionysis Zindros. “Proof-of-Stake Sidechains.” To appear at IEEE S&P 2019.