I think there is a lot discussion of extermination efforts here, which are not really relevant to the topic at hand which discusses what will happen with species if they remain in situ, not whether or not they should.
The dingo originated as an invasive species, it's just been there for long enough that you can't tell so much because the way the ecosystem worked before it got there has never been seen, and because it now fills the rolls filled in the past by marsupial predator, whether or not it directly influenced their declines. I don't consider the dingo native, per se, but it is naturalized and its place in the ecosystem that remains is stable, and it should not be removed. It almost certainly would not have been beneficial when it was first introduced.
It will be the same in another ten thousand years with whatever new species of placentals survive that long. It doesn't take too long, really, for animals to start fitting more "neatly" into the ecosystem after an initial destructive period where competitors lack adaptations to compete with or defend against them. Putting aside entirely the ethics of extermination efforts entirely, I consider it unlikely that certain species like the house cat can ever truly be removed from the continent of Australia. They are too small, cryptic, and over a vast and little-inhabited region, and they can continue to be released through pet owners unless very dramatic laws are enacted that reduce their prevalence in captivity. Let's assume in this example that this is the case, and cats remain there as dingo ancestors did. Eventually, after a lot of extinctions of smaller prey animals that don't fare well against it, it will become like the dingo, and those which remain will have adapted survival strategies that work, and things will go on.
There are a lot of extinctions in the short term regarding introduced species, but also rapid behavioral adaptations among the initial survivors. A few birds and rodents in Australia have already learned to safely kill and eat cane toads, a novel toxic species, in less than 90 years. That is literally nothing as far as evolutionary time scales go, which usually operate on hundreds of thousands to millions of years. Invasive is an arbitrary term that eventually can't keep applying to a species. When is the cut off? A hundred years? A thousand? A million or more? Tens of millions of years ago, monkeys colonized South America from an ocean crossing across the Atlantic and if they were like any modern mammal introduction, likely contributed to the extinction of some other species before evolving into new forms that could not be considered anything but native to this new region. Australia has received a very large number of adaptable, generalist animals all at once through anthropogenic means, but animals have been reaching new regions and altering ecology and contributing to the extinction of other animals as long as animals have existed, and I don't consider this inherently unnatural. There is a time limit on how quickly to remove these animals from their new ecosystems before they become a more intrinsic part of them - because eventually, they will do so, and the ecosystem will reach a new norm, and over million of years remaining animal life will evolve into new species in a more stable ecology.
The dingo is an interesting example because it is one of the only species to have reached that point, where removing it is now directly harmful to the remaining ecosystem, and where prey species have effective survival strategies against canid predators. As awful as the concept of a future Australia where cats remained at large killing native wildlife may sound, in a few thousand years the cat would probably be much less destructive to the ecosystem than it is now, only 200 years after its introduction, and species which survive this early period would become less vulnerable. Australian cats are experiencing rapid phenotype changes as the outback is a distinct ecosystem with novel evolutionary pressures. Cats that live there for long enough will speciate into new forms, and if any remain in ten thousand years and there are humans left to see them, these new subspecies of cats could not really be considered invasive anymore. Left for millions of years, they would give rise to an Oceanian radiation of felid species, and then new genera, until they were no more introduced than the new world monkeys. It will come at the loss of previous biodiversity, and this is where people oppose the very idea, but on timescales larger than humans are good at imagining, life will still endure and become something new in the aftermath. Dinosaurs would find the idea of a world overrun with megafaunal mammals terrible if they had the capacity, but their extinction allowed the evolution of billions of new organisms that would never have existed otherwise over the following 66 million years. On long time spans events such as this are major motivators of evolutionary changes.
