After now spending a few months having switched from DSLR to Mirrorless, I thought I'd provide my experience to this question.
Bokeh / DOF (M43)
The most noticeable difference I have found was shallow DOF. Having switched from a 1.6x crop to a 2x crop without having wider apertures available, I'm now using compression to accomplish the same DOF.
As an example, I previous used a 50mm f/1.4 on a 1.6x crop body, giving me the equivalent of 80mm f/2.24. Looking through my images, I found myself averaging f/2.0 with this lens. After having switched, I'm now using a 75mm f/1.8 and yielding slightly better results in areas I'm able to back further away from the subject. In tight quarters, I'm simply not as able to accomplish the same DOF, but I'm getting enough subject separation to not distract the viewer with the background.
Note: I say M43 because there are Mirrorless FF cameras available where this issue does not apply.
The largest disadvantage to mirrorless is in lower light when the camera doesn't know which direction to start focusing and it starts in the wrong direction. While it's only a split second, it's enough to miss the shot if you are photographing fast action. I have not felt disadvantaged by this when doing portraiture.
Handling / Build
Mirrorless cameras, like DSLR cameras, come in a wide variety of sizes, builds, and UI differences. While this is preference, I found a better selection of handling and UI with the mirrorless selection. While it is a smaller body to work with, the button placement feels very natural and intuitive.
Mirrorless build offers the same weather sealing as many of the top brands. I have tested my mirrorless camera in the rain and extreme cold without issues. Unless you plan to spend $6k+ on a DSLR, the same build quality is available between DSLR and mirrorless.
This is a hot topic for many photographers as the viewfinder is electronic (EVF) as opposed to optical. I was very uncertain about the changes, but have found it to be a improvement for my uses.
One of my favorite features as a portrait photographer is being able to see real-time exposure compensation in the viewfinder. You can expose to the subjects face and watch your exposure adjustments in real time. You can also view the histogram in the EVF before you take the shot. During portrait shoots, I also have a preview appear in the EVF without having to review on the back of the screen. This saves time during the shoot, reduces chimping, and reduces post production adjustments as well.
I haven't found any issues with low-light as the camera boosts the image in the EVF to assist composition.
Having focus peaking or magnification in the viewfinder has resulted in less trips back and forth between the back panel and viewfinder. I'm also able to preview an image in the EVF on very bright days instead of covering the back screen to be able to see the image.
Lens / 3rd party Accessory Selection
This depends heavily on the brand of mirrorless. If you move toward the M43 mirrorless brand, you will find a large selection of lenses in amateur and pro category. I have found that these lenses I need are all available in professional quality. As a portrait photographer, the M43 75mm 1.8 and 45mm 1.8 are an excellent pair.
I wouldn't be able to speak for 3rd party accessories as I no longer have a need for them. There are less available, but you would need to consult your practice and the ability to meet your needs.
Much like DSLR cameras the quality of the camera dictates the features available, but mirrorless is able to optically control up to 4 groups of strobes remotely from the camera body. This was greater than the 3 groups available on my Canon 7D.
M43 seems to peak at 16MP. Due to sensor size, you will be limited. With high-end glass, you're able to crop images and still maintain print quality. Compared to APS-C bodies, you will yield similar results in print, but have less ability to crop if needed.
A very interesting topic has been professional appearance with a smaller camera. While I use one of the larger mirrorless bodies, it is still significantly smaller than DSLR bodies and the lenses are much smaller. The only noticeable difference is that my clients appear to be actually more comfortable in front of the camera. They are able to see more of my face, my emotions, smile, etc. and it reduces the "social distance" between the client and the photographer. Clients are also less concerned about people's reaction in public as well.
In times that I need to appear more professional, like an event, I keep a staff badge in my backpack. This keeps others from jumping out in front of me or glaring when I'm somewhere spectators shouldn't be. Another helpful piece to garner more respect is that I sometimes still use my large DSLR backpack that gives the appearance of a lot of equipment. No one needs to know that it's only half full and 1/4 the weight.
After having switched for a few months, I'm overall very pleased with the decision. While I still gripe about the lacking DOF, I'm able to accomplish the DOF that fits my style. This may not be the same case for you and I cannot stress this enough. It is a gap that cannot be closed (DOF joke intended).
As for the size, weight, EVF, lens selection, professional features, etc. I'm very pleased.