A mirrorless camera consumes more power mostly because the circuitry is constantly running. Both the sensor and EVF or LCD have to be powered continuously in order to maintain the Live-View which is necessary for framing.
In contrast, a DSLR can even be used to frame while powered off. The viewfinder requires no power at all and the status line below the view is made of segmented displays which consume virtually no power. An EVF on the other hand with millions of pixels requires much more power to operate and refresh at a sufficiently high rate.
For the EVF or LCD to be fed, the sensor which is essentially a fairly large chip must be also read at a sufficient rate. This is highly power consuming. The sensor on a DSLR on the other hand only needs to be powered to read the exposure off of it.
When you use Live-View on a DSLR, it essentially becomes a large mirrorless, keeping the sensor on constantly and refreshing the LCD screen. Even though, mirrorless are optimized for this case (many DSLR will turn off Live-View after a few minutes or when the sensor reaches a certain temperature), it is often the case that a mirrorless uses a smaller less power battery. This means you will likely suffer even more from battery-life with a mirrorless than with your DSLR.