We study the cosmological evolution of extended branes in 6D warped flux compactification models. The branes are endowed with the three ordinary spatial dimensions, which are assumed to be homogeneous and isotropic, as well as an internal extra dimension compactified on a circle. We embed these codimension 1 branes in a static bulk 6D spacetime, whose geometry is a solution of 6D Einstein-Maxwell or Einstein-Maxwell-dilaton theories, corresponding to a warped flux compactification. The brane matter consists of a complex scalar field which is coupled to the bulk U(1) gauge field. In both models, we show that there is critical point which the brane cannot cross as it moves in the bulk. We study the cosmological behaviour, especially when the brane approaches this critical point or one of the two conical singularities.