Here's what I've gathered in my years of hunting in relation to the moon.....

1.) During a New Moon, or if the moon has been completely obscured by cloud cover all night, deer seem to move a lot in the early morning.

2.) When the moon is greater than half full and casts substantual light during the night, deer will feed at night and bed down in the morning. Most movement seems to happen around lunch time. However, you may catch deer in the mornings on their way to their beddeing areas.

3.) When the sun and the moon share the same sky during the day, you better be up in your tree stands, because deer will be on the move.

These are behaviors that I have personally observed, and I hunt according to what the moon is doing.