  We investigate the complete family of (aligned) Robinson-Trautman spacetimes
sourced by conformally invariant non-linear electrodynamics in $D$ dimensions
in the presence of an arbitrary cosmological constant. After presenting general
features of the solutions (which exist only in even dimensions), we discuss in
more detail some particular subclasses. Static metrics contain dyonic black
holes with various possible horizon geometries (K\"ahler if there is a magnetic
field, including flat branes) and different asymptotics. In addition, there
exist also time-dependent solutions (not possible in the $D>4$ linear theory)
which may represent white hole evaporation by emission of electromagnetic
radiation (or a time-reversed picture of black hole formation). For those, we
comment on a quasi-local characterization of possible past horizons. Finally,
we briefly discuss the special case of stealth solutions. In an appendix, a
theory-independent result on the redundancy of the gravity part of the field
equations for Robinson-Trautman spacetimes is further obtained.
