  We study homological and homotopical aspects of Gorenstein flat modules over
a ring with respect to a duality pair $(\mathcal{L,A})$. These modules are
defined as cycles of exact chain complexes with components in $\mathcal{L}$
which remain exact after tensoring by objects in $\mathcal{A} \cap
{}^\perp\mathcal{A} = \mathcal{A} \cap \Big( \bigcap_{i \in \mathbb{Z}_{> 0}}
{\rm Ker}({\rm Ext}^i_{R^{\rm o}}(-,\mathcal{A})) \Big)$. In the case where
$(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that
$\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$,
$(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half
of a hereditary complete cotorsion pair) we prove that these relative
Gorenstein flat modules are closed under extensions, and that the corresponding
Gorenstein flat dimension is well behaved in the sense that it recovers many of
the properties and characterizations of its (absolute) Gorenstein flat
counterpart (for instance, it can be described in terms of torsion functors).
The latter in turn is a consequence of a Pontryagin duality relation that we
show between these relative Gorenstein flat modules and certain Gorenstein
injective modules relative to $\mathcal{A}$. We also find several hereditary
and cofibrantly generated abelian model structures from these Gorenstein flat
modules and complexes relative to $(\mathcal{L,A})$. At the level of chain
complexes, we find three recollements between the homotopy categories of these
model structures, along with several derived adjunctions connecting these
recollements.
