[DG]

"draws BOB, heads to first and as he is approaching first the coach at third base box yells across diamond for a certain player to run as courtesy/speed up runner. So the batter-runner turns and heads toward dugout"

I read the original post to mean the batter did not reach 1B. I can not read the post any other way. The batter "draws a BOB", so the ball is LIVE. A base on balls is considered an award of 1B, but the ball is LIVE during the award. How many times has a defensive coach requested time while a batter walks to 1B and you say "not till he reaches 1b"? This was not an intentional walk in which case the ball is dead for the award.

The batter does not have to reach DBT to be declared out for desertion, just far enough for the umpire to believe he is deserting.