The invention relates to machines and methods for separative machining of a plate-shaped workpiece by a processing beam. The machines include a first movement unit configured to move the workpiece in a first direction and a second movement unit configured to move a machining head configured to emit the processing beam. The second movement unit is configured to move the machining head in a second direction perpendicular to the first direction to direct the processing beam onto the workpiece. The machines include a first workpiece support unit including a first workpiece-bearing face and a second workpiece support unit including a second workpiece-bearing face spaced apart by a gap from the first workpiece support unit and the first workpiece-bearing face. The gap extends along the second direction. The machines include at least two support slides configured to move in the gap in the second direction mutually independent of one another.