  In the context of gravitational theories describing renormalization group
flows across dimensions via AdS/CFT, we study the role of higher-derivative
corrections to Einstein gravity. We use the Null Energy Condition to derive
monotonicity properties of candidate holographic central charges formed by
combinations of metric functions. We also implement an entropic approach to the
characterization of the four-derivative flows using the Jacobson-Myers
functional and demonstrate, under reasonable conditions, monotonicity of
certain terms in the entanglement entropy via the appropriate generalization of
the Ryu-Takayanagi prescription. In particular, we show that any flow from a
higher dimensional theory to a holographic CFT$_2$ satisfies a type of
monotonicity. We also uncover direct relations between NEC-motivated and
entropic central charges.
