DISCLAIMER: I'm happy to be educated if I've got this wrong. I'm extremely sensitive to the fact that this is complicated, changing really fast, and I could easily be fully full of shit.
But as far as I understand it's all some variation on "we're going to anchor something on the slow, expensive, trustworthy chain that represents a promise that we're going to settle stuff over here and report back to the big, slow, expensive but trusted chain".
Often that anchor is some variation of a multi-sig custodial wallet with a bunch of blockchain bigshots on it, i.e. a fucking bank. If people want to evade banking regulations that's their business, but calling it cryptographic novelty is stupid.
Sometimes it's "we're going to settle transactions on some shard, so it doesn't matter that it's somewhere between a slow wristwatch and a fast potato in terms of throughput", in which case the people I want to transact with need to participate in that shard, in which case you've got Metcalfe's Law working against you: you're on the wrong side of the polynomial. I want to buy some shit from merchant $FOO but they're not on my shard. Damn, guess I need to route some $COIN up through the real chain and down to the shard where my pot dealer is.
The point is that if the thing you were deferring the transaction onto was both trustworthy and scalable, then you'd just use that. It's this non-terminating recursion where each invocation of the Y-combinator kicks off a fat token offering. Which is why the Lightening whitepaper is now $ZILLION pages long when it started at 18: every hole they patch creates 3 more.
IO-HK is somewhere south of RenTech and somewhere north of NASA in terms of the Nobel laureates they've got working on Ouroboros [BFT|Praus|Genesis]? and they can't get it to work: ADA is getting pushed around a warmed-over Tendermint still (really fast potato).
The ParityTech people seem to have learned the hard way that Google barely made it fly with friggin cesium clocks and satellites in a non-Byzantine setting when they put together BABE [BADASS|SASAFRASS|WHATEVER]? and slapped GRANDPA on it. Ethereum v2 gets delayed with roughly the same cadence as viable fusion power (but Casper the FFG!).
If someone knows how to do a chain that's truly secure, truly decentralized, and doesn't cost $500 to buy a $5 ENS domain then I missed that paper.
If you know of a project that actually works I want in on the governance token. It'll make BTC look like a victim of hyperinflation.