The arena tiers in classic arena change instantly when you get the points to climb up or, by losing fights, go down. If you win a fight that brings you into Bronze 2, you are at the very bottom of that tier. If your arena defense takes a loss, you will moved down to Bronze 1 again, but on the top of that tier. One victory will take you to Bronze 2 again, and so on. If you reach a new tier, try to win some more fights to get a bolster that may take a few losses from your defense.
In addition to that, once a week (Monday, I think at 8:00 a.m. UTC) the arena tiers will be checked and the reward chests are given. On that point, everybody will lose all surplus arena points and be moved to the bottom of their actual arena tier.