It is known that islands that are closer to other land where species disperse from end up with higher biodiversity, other factors being equal. This website
on island biogeography sums it up with this quote,
A modified version of the classical island biogeography model proposed by MacArthur and Wilson (1963) is depicted above. The model considers the interaction of two main parameters, colonization and extinction, and then considers island size and distance from mainland as predictors of the species richness found on each island. Both colonization and extinction can be thought of in terms of rates or probabilities, and the size and isolation of islands impacts the probability of colonization and of extinction. Simply, Islands close to the mainland will more readily receive colonists from that mainland and, similarly, probability of colonization of large islands will be higher than that of small islands. Thus the larger the island and closer to the mainland, the more potential species will arrive. Species will also more readily go extinct on small islands than large, due to factors such as smaller population sizes and less available habitat. These parameters go hand in hand predicting that species richness will peak in large islands near mainlands, and be lowest in small islands far away from the mainland.
I have seen a few aggressive invasive species that do decrease biodiversity in the patches they take over, although oftentimes the claims are exaggerated (I've seen areas labeled monocultures of an exotic species that I can see quite a few other species growing in). However, we are glimpsing a very small period of time, as well as one with a lot of other human-induced factors that are changing the environment, so its no surprise that different species are thriving in the changed conditions. Humans have greatly accelerated the movement of species around the world, but there has always been a certain amount of movement, and I think the above example of island biogeography shows that more migration tends to increase biodiversity overall. That doesn't mean that none of the new species caused the extinction of species that were previously present, but the number driven extinct must end up fewer than the number of arrivals.
I've heard from some nativists the idea that certain exotic species will never find a balance and simply take over and leave much of the planet as near monocultures. If that were true, I'd think the islands with more migrating species would end up with less biodiversity, as periodically an aggressive species would come through and wipe out most of the others. Herbicide companies do love it when people have this view of exotic species, and have been known to support native plant groups and push them toward chemical use to eradicate exotics.
There was a time in early American history when peach trees were aggressive enough on parts of the east coast that some people worried they were taking over the landscape in places. That was before pests and diseases caught up with them, some imported themselves but others adapting from native wild stone fruits. Now there are occasional naturalized peaches but they don't appear on any invasive species list. I imagine the same will happen eventually to the invasives of most concern today, more pests and diseases will come after them and balance the populations out, how long that will take is the big question, my guess is that species that have native relatives will have more pests/diseases move to the new species more quickly.
While I'm generally not in the nativist camp, there are certain aggressive species that I would never spread to new areas, because even if they will eventually come into balance, in the meantime I wouldn't want to chance it by adding one more stress to all the others going on in this era.