Inside a budget's rules, a payment leaves the DAO in seconds. Anything else becomes a normal council vote. The money stays in the DAO until a payment goes out.
This page reads live data from NEAR testnet: the demo DAO harbor-builders.sputnikv2.testnet, its three budgets on the Petty Cash contract pettycash.testnet, every payment and every council vote. You act as a budget holder: request a payment, and inside the rules it is paid from the DAO in seconds; outside the rules it goes to a council vote, which you can also cast as a demo council member.