  The discovery of GRB 170817A, the first unambiguous off-axis short gamma-ray
burst arising from a neutron star merger, has challenged our understanding of
the angular structure of relativistic jets. Studies of the jet propagation
usually assume that the jet is ejected from the central engine with a top-hat
structure and its final structure, which determines the observed light curve
and spectra, is primarily regulated by the interaction with the nearby
environment. However, jets are expected to be produced with a structure that is
more complex than a simple top-hat, as shown by global accretion simulations.
We present numerical simulations of short GRBs launched with a wide range of
initial structures, durations and luminosities. We follow the jet interaction
with the merger remnant wind and compute its final structure at distances
$\gtrsim 10^{11}$~cm from the central engine. We show that the final jet
structure, as well as the resulting afterglow emission, depend strongly on the
initial structure of the jet, its luminosity and duration. While the initial
structure at the jet is preserved for long-lasting SGRBs, it is strongly
modified for jets barely making their way through the wind. This illustrates
the importance of combining the results of global simulations with propagation
studies in order to better predict the expected afterglow signatures from
neutron star mergers. Structured jets provide a reasonable description of the
GRB 170817A afterglow emission with an off-axis angle $\theta_{\rm obs} \approx
22.5^\circ$.
