  The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek,
Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a
subgraph of the strong product of a graph with bounded treewidth and a path.
This result has been the key tool to resolve important open problems regarding
queue layouts, nonrepetitive colourings, centered colourings, and adjacency
labelling schemes. In this paper, we extend this line of research by utilizing
shallow minors to prove analogous product structure theorems for several beyond
planar graph classes. The key observation that drives our work is that many
beyond planar graphs can be described as a shallow minor of the strong product
of a planar graph with a small complete graph. In particular, we show that
powers of planar graphs, $k$-planar, $(k,p)$-cluster planar, fan-planar and
$k$-fan-bundle planar graphs have such a shallow-minor structure. Using a
combination of old and new results, we deduce that these classes have bounded
queue-number, bounded nonrepetitive chromatic number, polynomial $p$-centred
chromatic numbers, linear strong colouring numbers, and cubic weak colouring
numbers. In addition, we show that $k$-gap planar graphs have at least
exponential local treewidth and, as a consequence, cannot be described as a
subgraph of the strong product of a graph with bounded treewidth and a path.
