Methods and systems for continuous multistage processing of particulate solids that can control the distribution of residence times of processed particles. One disclosed method includes admitting fluidizing gas into transfer tunnels between fluidized bed vessel stages, using gas inlets located in or outputting directly into the transfer tunnel to promote flow of the fluidized solids through the transfer tunnel. A common gas distributor plate can supply gas to the stage vessels and to any transfer tunnels having gas inlets. The distributor plate can be separable from the system or openable to provide access for removal of obstructions and for maintenance. A common base plate can incorporate lower portions of the stage vessels and the sides and roofs of the transfer tunnels, the bottoms of the vessels and tunnels optionally being provided by the gas distributor plate. Some multistage systems and processing methods have from 15 to 200 or more stages.