  We construct the Carrollian equivalent of the relativistic energy--momentum
tensor, based on variation of the action with respect to the elementary fields
of the Carrollian geometry. We prove that, exactly like in the relativistic
case, it satisfies conservation equations that are imposed by general
Carrollian covariance. In the flat case we recover the usual non-symmetric
energy--momentum tensor obtained using N\oe ther procedure. We show how
Carrollian conservation equations emerge taking the ultra-relativistic limit of
the relativistic ones. We introduce Carrollian Killing vectors and build
associated conserved charges. We finally apply our results to asymptotically
flat gravity, where we interpret the boundary equations of motion as
ultra-relativistic Carrollian conservation laws, and observe that the surface
charges obtained through covariant phase-space formalism match the ones we
defined earlier.
