The domain wall nucleation and motion processes in Permalloy nanowires with a thickness gradient along the nanowire axis have been studied. Nanowires with widths, w = 250 nm to 3 um and a base thickness of t = 10 nm were fabricated by electron-beam lithography. The magnetization hysteresis loops measured on individual nanowires are compared to corresponding nanowires without a thickness gradient. The Hc vs. t/w curves of wires with and without a thickness gradient are discussed and compared to micromagnetic simulations. We find a metastability regime at values of w, where a transformation from transverse to vortex domain wall type is expected.