I'll try to give a practical explanation based on my own experience. The mind is not an object or entity. In fact, objects and entities don't exist. Reality is made up of moments of experience. Moments of hearing, seeing, smelling, tasting, touching, and thinking. What we call the mind is just a part of these moments of experience. Every moment a new mind arises and passes away. As we fall asleep a series of experiences arise and pass away: drowsiness, pleasure, thinking, etc. After these experiences cease the next set of experiences that arise are either dreams or waking up.
There is nothing that stops existing when we fall asleep or when we die because there was nothing there other than these moments of experience that are constantly arising and passing away.
I hope that helps clarify some things for you.