The global dynamics of traveling-wave patterns in convection in a mixture of ethanol in water is studied in different cell geometries: circular, rectangular, and stadium-shaped cells. The dynamics in these cells differ greatly, changing from a globally rotating state in the circular cell, to one large domain of locally parallel traveling waves in the rectangular cell, to a continually chaotic state in the stadium cell. In all three cases, the patterns can be described in terms of the phase of the complex order parameter. Disorder in the patterns is quantified in terms of topological defects in the phase field. While the numbers, net charge, and dynamics of defects differ greatly in the patterns in the three cells, the local dynamics of the defects, as measured by the defect-defect correlation functions, are similar.