Mirrorless cameras come in smaller bodies and can come with smaller sensors as well. Sony has a full frame mirrorless. Fuji has some APS-C size sensors, and then there is an array of micro-four thirds cameras, such as Olympus OMD line.
Everything is a tradeoff, but the smaller sensors let you use smaller lenses. Some of those lenses are incredibly good.
The fuji with APS-C sensor is well regarded as having image quality as good as a DSLR. The Olympus OMD EM1 is considered a high performer as well.
Typically, you'll see poorer low light performance with smaller sensors. However, the Olympus OMD EM1 is an excellent performer in the 95% case range for up to the advanced enthusiast.
I'm using the Olympus OMD EM1 with the Olympus 12-40 f/2.8 and it produces excellent images.
Samples (Olympus OMD EM1 with 12-40 2.8)
With the larger sensors, you get more shallow depth, because with a crop sensor, you need to be at a greater distance to achieve the same field of view, so if you're used to the DOF of an f/2.8 on a full frame, you'll need an f/1.4 to get the same DOF on a micro four thirds camera.
Other considerations are that the current MFT cameras aren't necessarily as good as DSLRs in the autofocus department, although the Oly OMD EM1 is remarkably fast and accurate. It's the perfect camera for me.
I used to carry a Canon 5D mk iii with L lenses. I sold all that gear for the OMD EM1 and have never looked back.