As a general term, sinking mind denotes any dimming of awareness. At its best, it is sort of a mental vacuum in which there is no thought, no observation of the breath, no awareness of anything. It is a gap, a formless mental gray area rather like a dreamless sleep. Sinking mind is a void. Vipassana meditation is an active function.
Concentration is a strong, energetic attention to one single item.
Awareness is a bright clean alertness. And sinking mind contains neither. At its worst, it will put you to sleep. Even at its best it will simply waste your time. When you find you have fallen into a state of sinking mind, just note the fact and return your attention to the sensation of breathing.