The Reynolds transport theorem provides a generalised conservation law for any conserved quantity carried by fluid flow through a continuous domain, and underpins all integral and differential analyses of flow systems. It is also intimately linked to the Liouville equation for the conservation of a local probability density function (pdf), and to the Perron-Frobenius and Koopman evolution operators. All of these tools can be interpreted as continuous temporal maps between fluid elements or domains, connected by the integral curves (pathlines) described by a velocity vector field. We present new formulations of these theorems and operators in different spaces. These include (a) spatial maps between different positions in a time-independent flow field, connected by a velocity gradient tensor field, and (b) parametric maps -- expressed using an extended exterior calculus -- between different positions in a manifold, connected by a vector or tensor field. The analyses reveal the existence of multivariate continuous (Lie) symmetries induced by a vector or tensor field associated with a conserved quantity, which will be manifested in all subsidiary conservation laws such as the Navier-Stokes and energy equations. The analyses significantly expand the scope of methods for the reduction of fluid flow and dynamical systems.