Garden cultivars with overlarge flowers they can't support without staking. That's why it happens after they open and enlarge.

Jim,
You don't say if the stems bend just under the flowers, or if the entire stem falls down. If the stem bends under the flowers, this can be a fungal problem. If the entire stem is falling over, it could be weak because of too much fertilizer, or the shallow containers that you mention. 10 inches is pretty shallow for large annuals such as the ones you describe. Flowers that are grown in too much shade can also become too lanky and weak.

You could, perhaps, add another layer of rocks or timbers and build up your beds. Plant supports might help as well, and keep the plants on a lean diet.