A rotary filling machine for filling containers with bridgeable dry materials includes a turret supporting a plurality of circumferentially spaced drop buckets and a plurality of funnel assemblies located under the drop buckets. A stationary slide plate is located vertically between the funnel assemblies and the drop buckets. When viewed in a direction of turret rotation, the slide plate has an upstream end, a downstream end, and inner and outer edges. A portion of the slide plate is tapered progressively in diameter toward its downstream end such that flow paths from the bottoms of the drop buckets to the inlet openings of the funnel assemblies increase progressively in diameter with the taper of the slide plate. Also provided is a funnel assembly that dilates bridgeable materials before dispensing those materials into a container.