One way to argue for that would to be in terms of essences or natural kinds (which capture the essence) in regards to properties or things/objects/entities. Thus, even if peripheral aspects of entity/property X change (or peripheral stuff changes X), if “all things remaining equal”, then the essence of X will remain the same.
There are indeed what some people call “causal islands”: abstract objects such as numbers, propositions, etc. (If you accept such things.) In addition, if an object or a property has an essence, then that object will retain its essence “all other things remaining equal”. In other words, in this language, essences are indeed causal islands. They must also be independent. Indeed you can say that essences are also abstractions (if not exactly abstract objects).
Again, in the causal flux, object/property X will keep its essence.
Does the belief in the aforementioned “self-subsistent” (as it were) entities (or properties) involve an automatic philosophical commitment to (or assumption of) the phrase “all other things being equal”? That is, that property or entity X would have exactly this nature regardless of all else?
It can be said that if you change one thing, then you change everything else too. (As various kinds of monist have argued.) Or, less dramatically, if you change one thing, then you must automatically change other things.
For example, if President Obama had an extra finger, then there would also be many other differences too. Even if President Obama had an extra molecule, many other things would be different. For one, that extra molecule would automatically affect other molecules. And that means that one tiny change (a different molecule) would bring about many other changes.
This means that X is dependent on other things. X will also affect other things.
Finally, surely saying that “X is an independent property” can't mean that X could exist without anything else. Although it can be said that X could exist without some of the things it now coexists with.
(Does all the above work against the various monistic theories - and theories of interdependence - in which no entity or property can ever have a nature apart from – all? - other things?)