In this work, we perform creeping-flow simulations of upper-convected Maxwell and simplified Phan-Thien-Tanner fluids to study the purely-elastic steady bifurcation and transition to time-dependent flow in three-dimensional planar cross-slots. By analysing the flow in geometries with aspect ratios ranging from the near Hele-Shaw flow like limit, up to the very deep, two-dimensional limit, we are able to characterize the mechanism of the cross-slot bifurcation with significant detail. We conclude that the bifurcation mechanism is similar to a buckling instability, by which fluid is redirected via paths of least resistance, resulting in the emergence of peripheral stagnation points, above and below the central stagnation point. The intake of matter at the centre via the inlet axis is thus reduced, being compensated by fluid flowing through low resistance corridors along the central vertical axis, above and below the central point. Furthermore, we propose and locally compute a modified Pakdel-McKinley criterion, thereby producing a scalar stability field and suggesting emergent peripheral stagnation points also indirectly contribute to the onset of time-dependent flow. (c) 2015 The Authors. Published by Elsevier B.V.