I’ve had both DSLR and mirrorless kit out in similar temps on many occasions — yes keep batteries close to you whenever possible. Watch for lens fogging when gong between temp extremes.

Autofocus can also go a little wonky when the temps get real low as well — I’d keep them as warm as possible for as long as possible, ideally next to your body.

I went to northern Finland years ago and was shooting in temperatures down to -40, the advice I was given at the time was to keep the camera in its case to allow the temperature to equalise, particularly when going from cold to warm.

My normal routine was to go out, give the camera a good 10-20 minutes to cool down before shooting. Once finished at night I didn’t get the camera out at all once back in the hotel. I’d then sort out new memory cards and the like the next morning. If I changed a battery when I was out I kept it in a pocket so I could get at it to charge without opening my camera case.