This thesis addresses an ever-present problem in string theory — the top-down construction of lower-dimensional effective field theories (EFTs) of gravity. We tackle this through the use of two formalisms: 1. consistent truncations, and 2. inconsistent truncations.

Starting with consistent truncations, we present a construction of consistent embeddings of non-linear half-BPS supergravity theories on the worldvolumes of the M-theory M5-brane, the Type IIB D3-brane, and their diagonal and vertical descendants. Following this, we provide consistent truncations of Type IIA supergravity on a 5- and 6-dimensional manifold given by a $S^3$ twisted over a special Lagrangian (SL) 2- and 3-cycle, which capture the physics of NS5-branes wrapping the SL cycles in the near-horizon limit. We then provide a novel use of non-Abelian T-duality to construct new consistent truncations of Type IIA supergravity to the minimal gauged supergravity in 5-dimensions by T-dualising the SU(2) isometries in the $S^5$ and $Y^{p,q}$ truncations of Type IIB supergravity.

In the final parts of this thesis, we turn to inconsistent truncations by first developing a taxonomy of brane gravity localisation about a resolved NS5-brane with the Eguchi-Hanson space as its transverse space. A result of this taxonomy is the presence of a delayed covert form of symmetry breaking of the worldvolume diffeomorphisms when brane gravity localisation exists. This covert symmetry breaking is explored fully in a simpler model of scalar electrodynamics with one dimension restricted to a closed interval, on the opposing ends of which, the gauge potential obeys non-standard boundary conditions.