  In this paper, we study fan-planar drawings that use $h$ layers and are
proper, i.e., edges connect adjacent layers. We show that if the embedding of
the graph is fixed, then testing the existence of such drawings is
fixed-parameter tractable in $h$, via a reduction to a similar result for
planar graphs by Dujmovi\'{c} et al. If the embedding is not fixed, then we
give partial results for $h=2$: It was already known how to test existence of
fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how
to test this for trees. Along the way, we exhibit other interesting results for
graphs with a fan-planar proper $h$-layer drawings; in particular we bound
their pathwidth and show that they have a bar-1-visibility representation.
