There are a few different approaches to adding covenants and capabilities
to the UTXO model with varying tradeoffs. It turns out that it can be done
while making very few but not quite zero compromises to practices Bitcoin
has been following so far.

First, the good news: Full support for both capabilities and covenants can
be added without changing the UTXO model whatsoever by adding some more
programmatic capabilities to the language and doing some programmatic
tricks. Since scriptpubkeys/scriptsigs continue to run ephemerally at
validation time full turing completeness is much less dangerous than people
fear. The main thing missing from what's expressed in transactions
themselves is a coherent notion of a single parent of each output instead
of the all-inputs-lead-to-all-outputs approach of transactions currently.
It would also probably be a good idea to add in a bunch of special purpose
opcodes for making coherent statements about transactions since in Bitcoin
they're a very complex and hard to parse format.

Now for the controversial stuff. Once you start implementing complex
general purpose functionality it tends to get very expensive very fast and
is likely impractical unless there's a way to compress or at least
de-duplicate snippets of code which are repeated on chain. Currently
Bitcoin has a strong policy that deciding which transactions to let into a
block for maximum fee is a strictly linear optimization problem and while
it's possible to keep things mostly that way making it completely strict is
unlikely to workable. About as close as you can get is to make it so that
each block can reference code snippets in previous blocks for
deduplication, so at least the optimization is linear for each block by
itself.

Having covenants and capabilities at all is controversial in and of itself.
With covenants the main issue is whether they're opt-in or opt-out. For a
payment to someone to come with a rider where they could accept it and
think their system was working properly for a while until you exercised
some kind of retroactive veto on new action or even clawback would
obviously be unacceptable behavior. But for payments to come with covenants
but the recipient not even be able to parse them unless they're fully
buying into that behavior is much more reasonable.

The main issue which people have raised with capabilities is that if you
were to have colored coins whose value was substantially greater than the
chain they were tokenized on then that could potentially create a business
model for attacking the underlying chain. While this is a real concern
tokenized assets have been out for a while now and have never come close to
causing this to happen, so maybe people aren't so worried about it now.

Given all the above caveats it turns out one weird trick is all you need to
support general purpose capabilities: for a UTXO to have a capability its
scriptpubkey asserts that its parent must either be the originator of that
capability or also conform to the same parent-asserting format. More
complex functionality such as supporting on-chain verifiable colored coins
can also be done but it follows the same pattern: Capabilities are
implemented as backwards pointing covenants.

If you'd like to see a fleshed out implementation of these ideas (albeit in
a slightly different model) there's quite a bit of stuff on chialisp.com