I have never been in the Miami war room on Draft Day but here is what I think occurs. I welcome any comments if you think I'm wrong.
My read is that Chris Grier is the man designated in charge, coordinating Miami's Draft. However, they work as a team. Grier has put together the Draft board based on all the assessments and information collected by his team of scouts and experts and he will make the recommendation for each selection based on what the board is showing. It will be discussed with emphasis placed on needs. The question will be asked to Coach Gase if he has any objection and then each card will be submitted.
Grier, Tannenbaum, Gase and the entire player evaluation team have already discussed the merits of the players to derive the values assessed on our board. Tannenbaum is the top negotiator. If evaluating whether to move up or down, he is probably active in the negotiation with the other team(s) and Grier defers to him. There is an art to successful negotiation and very few are really good at it. However, make no mistake, Mike Tannenbaum is a pro, but he needs Grier's team's accurate assessment of player values. If our Draft board is wildly wrong, then it will lead to bad deals and poor decisions (e.g. Jeff Ireland moving up to take Dion Jordan in 2013).
Comments by owner Stephen Ross and others strongly suggest that Tannenbaum, Grier and Gase work well as a team. Mr Ross has even said that this is the most coordinated that the leaders have worked together since he bought the team. The benefit of this will only be proven over a period of time but there is scope for optimism.