  We construct new families of $\mathcal{N}=(0,4)$ $\mathrm{AdS}_3\times S^2
\times \tilde S^2 \times S^1$ backgrounds fibred over a 2d Riemann surface in
Type IIB string theory. These solutions are obtained by extracting the
near-horizon limit of D3-brane box configurations, consisting on D3-D5-NS5
branes ending on D5'-NS5' background branes. We relate our families of
solutions to previous $\mathrm{AdS}_3\times S^2 \times \text{CY}_2$ and
$\mathrm{AdS}_3\times S^3 \times S^2$ solutions to Type IIA recently
constructed in the literature. We construct explicit 2d quiver CFTs associated
to the D3-brane box configurations, and check that the central charges match
the holographic result. We extend our solutions to include D7-branes, and show
that a subclass of these solutions can be interpreted in terms of D3-D5-NS5
defect branes embedded in a 5d fixed point theory. This is explicitly realised
by linking our solutions to a 6d domain wall that asymptotes locally the
$\mathrm{AdS}_6\times S^2 \times \Sigma_2$ solution T-dual to the Brandhuber-Oz
vacuum.
