Motivated by finite element spaces used for representation of temperature in the compatible fi-
nite element approach for numerical weather prediction, we introduce locally bounded transport
schemes for (partially-)continuous finite element spaces. The underlying high-order transport
scheme is constructed by injecting the partially-continuous field into an embedding discontinuous
finite element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and projecting
back into the partially-continuous space; we call this an embedded DG transport scheme. We
prove that this scheme is stable in L
2 provided that the underlying upwind DG scheme is. We
then provide a framework for applying limiters for embedded DG transport schemes. Standard
DG limiters are applied during the underlying DG scheme. We introduce a new localised form of
element-based flux-correction which we apply to limiting the projection back into the partiallycontinuous
space, so that the whole transport scheme is bounded. We provide details in the specific
case of tensor-product finite element spaces on wedge elements that are discontinuous P1/Q1 in
the horizontal and continuous P2 in the vertical. The framework is illustrated with numerical
tests.