The director does not deal with money. That's the producer's job.
The producer works for the company fronting that budget. They are in charge of making sure the money is spent in the way intended. The director is the creative force of the film. So the director goes to the producer and says "we need all this shit to shoot this scene". The producer determines how much all that will cost. They do the same scene by scene by scene, and the producer writes a budget and a production schedule. Then that budget is presented to the producer's boss, the executive producer.
The executive producer looks at the budget and decides if he thinks the movie is worth it. Then he approves the budget (sometime with changes, e.g. "Do we really need this?" and argues with the director about why X is necessary and if they can get away with using cheaper Z"). The process takes a *long time* and movies can get stuck in "development hell" where the director, producer, and executive producer are just going back and forth forever.
There is not a lump sum transferred to the producer. Rather, every cost is given to the producer as an invoice, and it is billed receipt by receipt to the executive producer's company.
I'm not sure a single film has reached a $1B budget.
A film that large would be financed by a major studio. They set up a corporation just for that one film, and they loan the new corporation the money. They don't send it all at once; it's paid as needed.
Before the studio agrees to make the movie, the producers make a schedule and a budget from the script. This accounts for everything from how many days they need the actors, to how many lunches needs to be prepared, and how many special effects shots.
The process of budgeting and scheduling a huge movie takes a large team of *line producers* and *unit production managers* these are producers who do the nuts and bolts of the production. They are like sargeants -- they make the thing happen. In the process of making the schedule and budget, they will have to bid and consult with every department, from camera and electrical to editing and VFX. They will also negotiate deals with support providers like hotels and caterers.
When the budget and schedule are finalized, the studio may ask for revisions if they think it costs too much.
Some producers negotiate a bonus if they deliver under the budget. Some studios won't do this, because they want the best movie, so they can make more money from it.
If the film is radically under budget, it's seen as a failing of the producers, because they failed to estimate properly.
Look more closely into the role a Producer plays in the film process. ELI5, the producer is essentially the CEO of the film. The director is the top **creative** talent on the film who is, truthfully, just another hired hand like anyone else working on the movie. A producer's core responsibilities are to handle the entire production process including hiring, budgeting, scheduling, and.... what it really evolves into.... the lead firefighter who manages every major crisis that comes up. Major films are largely financed by the large studios. Those studios will hire (or appoint) a Producer to manage the production (be the movie's CEO). Executive Producers are typically appointed by the studio to act as the eyes and ears of the studio executives. They will typically be the most conservative voices in terms of spending money. Costs are typically paid to vendors from the studio.