In this work we present an extension of a front-tracking algorithm with a dynamical redistribution of surface markers for interface advection of a fluid body in unstructured quadrangular grids. The interface is described by an ordered list of markers connected by segments which are first advected along the streamlines and then redistributed uniformly along the interface while conserving the spanned area. In this article we perform two-dimensional tests over unstructured meshes and compare the results with those obtained in Cartesian grids. Solid body motions, such as translations and rotations, and more complex vortical flows which progressively deform and stretch the interface line are considered over different geometries. The results show that the interface evolution can be tracked successfully in challenging situations with a very good accuracy in terms of area conservation and a rather moderate decrease in the performance of the method with respect to the structured Cartesian mesh.