Surprise: the question has two sub-answers that are needed before we get to the final answer.
The first sub-answer is that mirrorless has already replaced DSLRs for some people. Well, not quite "replaced." Let's say supplemented, as in mirrorless is now a legitimate alternative to DSLRs for a number of uses. That's certainly true for casual shooting, static types of photography such as landscapes, travel photography, portraits, street photography, and even some photojournalism and event photography.
The primary ingredient that allowed mirrorless to be considered over DSLRs in these situations tends to center around size and weight. Most mirrorless cameras simply create smaller and lighter shooting kits than DSLRs, so as the user base aged and also got tired of carrying big, heavy bags of gear, they started considering the mirrorless alternatives.
What they found in 2009 is a lot different than what they find today. In 2009 there likely wasn't a fast refresh and high dot count EVF, the focus system tended to meander to the focus point, the feature sets weren't complete, and the controls tended to be based on compact cameras and designed for the Japanese definition of beginners. Plus, there weren't many lenses available. Today we have fast and high resolution EVFs, fast focus systems, full feature sets, the controls are better designed and implemented, and several of the providers now have full (m4/3) or fairly full (Fujifilm X, Sony FE) lens sets.
In other words, mirrorless has achieved a level of parity with DSLRs on a number of fronts, while being less behind on others. Whether you consider mirrorless cameras as appropriate for your shooting depends upon whether parity was reached in the things you value. Or perhaps close-enough-to-parity but you value the smaller, lighter aspect.
But the second sub-answer is that mirrorless hasn't replaced DSLRs and hasn't yet fully reached parity with them, either. The primary types of photographer that tends to believe that answer is almost always either shooting sports or wildlife. In both these uses, it is lenses plus focus performance while tracking subjects that tend to come up short on the mirrorless side.
Yes, I know that the m4/3 world now has a very fast-focusing E-M1 Mark II plus a full set of lenses out to 300mm f/4 (600mm equivalent). But the truth of the matter is this: motion tracking results in continuous shooting still don't match those of the best DSLRs; you pay a penalty that you can't quite overcome for sports with the smaller sensor; and it's really only m4/3 that comes close to giving you useful long lens choice in mirrorless at the moment.
m4/3, Fujifilm X, and Sony FE come the closest to matching the full frame pro DSLRs for fast, erratic motion in low light, but each has missing pieces and none quite nail a motion sequence as reliably as the Canon 1DxII and Nikon D5, for instance. In particular, the big issues for each of the mirrorless contenders—smaller issues are present too—are:
m4/3 isn't a great low light choice
Fujifilm X is missing lenses
Sony FE hasn't reached focus speed/hit rate parity
Okay, so those are the sub-answers. What's the real answer? Simple: interchangeable lens cameras (ILC) come in a wide range of forms. They differ in sensor size, lens mount, performance aspects, and yes, whether or not they have an optical path with a mirror (DSLR) or not (mirrorless). There is no "sell by" date for DSLRs. I suspect that due to the large user base with substantive lens sets already purchased, DSLRs will continue to be around for quite some time, probably as long as mirrorless cameras are. Personally, I'm surprised that Canon and Nikon didn't drop size and weight faster out of their DSLR lineups, nor shore up their crop sensor lens lineups. Not doing that gave the other camera companies pursuing mirrorless essentially seven years to reach parity on other things, and now Canikon, particularly Nikon, finds itself with another competitive fight on their hands to maintain market share. One of the reasons why I started the sansmirror site six years ago was that I saw this parity in ILC cameras coming. Both design approaches have their pluses and minuses, and I think both will coexist for some time to come. If a mirrorless camera is a better choice for you, great, that's what this site (sansmirror) is for. If a DSLR camera is a better choice for you, great, that's what dslrbodies.com is for. I shoot with both types because there are some tasks I find one better suited for than the other, simple as that. I don't see this as a "one side wins" contest, I see it as "we all win." That said, remember that both mirrorless and DSLR cameras are part of systems. Systems are more than just camera bodies. Indeed, they're more than just one size camera body. Canon—and to a lesser degree Fujifilm and Sony—seems to get this, and Canon is the only company that currently has a mirrorless and DSLR system that's aligned fairly well and easily allows its user base to crossover as they see fit. I can only expect that to get better over time. Bottom line: buy what you need. Don't get hung up on the mirrorless/DSLR debate, as you'll still be debating that five (and more) years from now if you do.