Player positions and availability for a contest are set and locked at the time the draft begins.
What does this mean for your contests and rosters?
If an MLB player is set as a DH for his team when your draft begins, he will remain as a DH for the entirety of your contest, even if he is moved from DH to 3B, he will always remain in your DH roster position. This would be the same for NBA players who may move between the Center and Power Forward position.
When your draft begins, the only available players will be players signed to a contract and with their Teams active roster. This means that any player, at the time of the draft, who is assigned or with an affiliate team, will not be available for the draft, or to be added to your roster at a later time if they are called up.
i.e. MLB or NHL players called up from their farm teams. NFL players moved from the practice squad to the active team roster. Players who are unsigned at the time of your draft but sign a contract later that week. All these moves will not make those players available to be added through swaps or through the waiver wire during your contest.
If a player is dropped at anytime during a contest, that player will not be available to be added to any roster for the remainder of said contest.