The transition between non‐rifted continental lithosphere and oceanic lithosphere in rifted margins can display a wide range of characteristics, depending on the regional tectonic evolution. The velocity and duration of the rifting process as well as the geodynamic setting influence the properties and geometry of the margins, which are often grouped into two main categories: magma‐poor and magma‐rich. We show how different types of rifted margins can influence the dynamics of continental collision, focusing on the time and depth of slab break‐off after collision and the fate of margin material. We find that rifted margins have a noticeable impact on subduction dynamics, as we observe large variability in slab break‐off times and depths. In particular, the presence of a rifted margin can delay slab break‐off to up to 60 Myr after the onset of collision. Our results show that a large portion of the weak crust of magma‐poor margins is likely to detach from the subducting plate and accrete to the upper plate, while the dense and strong mafic and ultramafic component of magma‐rich margins causes most of the margin to subduct and be lost into the mantle, leaving only a small fraction of transitional and oceanic crust at the surface. Therefore, the volume of accreted material is much larger when the margin is magma‐poor than magma‐rich, which is consistent with geological observations that fossil magma‐poor rifted margins are preserved in many mountain ranges, whereas remnants of magma‐rich rifted margins are scarce.