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We develop Korevaar-Schoen's theory of directional energies for metric-valuedSobolev maps in the case of ${\sf RCD}$ source spaces; to do so we cruciallyrely on Ambrosio's concept of Regular Lagrangian Flow.  Our review of Korevaar-Schoen's spaces brings new (even in the smoothcategory) insights on some aspects of the theory, in particular concerning thenotion of `differential of a map along a vector field' and about theparallelogram identity for ${\sf CAT}(0)$ targets. To achieve these, one of theingredients we use is a new (even in the Euclidean setting) stability resultfor Regular Lagrangian Flows.

Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows