Can drawing a weapon be part of ANY kind of move action or just moving?

Assuming a character has a BAB higher than +1, would he be able to draw a weapon (or shield) as a move action and then draw (or sheath) another weapon (or shield) as part of that same move action (due to his BAB)?

This question could be applied to any other move action, does it have to be specifically about moving the character or a character could draw a weapon even while mounting a horse or picking up an object?

What about the Two-Weapon Fighting feat? Does that allow for drawing two weapons (or shields) as part of any move action, like standing up from prone (still with BAB higher than +1)?