A multi-stage product gas generation system converts a carbonaceous material, such as municipal solid waste, into a product gas which may subsequently be converted into a liquid fuel or other material. One or more reactors containing bed material may be used to conduct reactions to effect the conversions. Unreacted inert feedstock contaminants present in the carbonaceous material may be separated from bed material using a portion of the product gas. A heat transfer medium collecting heat from a reaction in one stage may be applied as a reactant input in another, earlier stage.