Let
K
K
be a closed convex cone in the state space
R
n
\mathbb {R}^n
. This note characterizes the
K
K
-monotonicity of a non-autonomous dynamical system
x
˙
(
t
)
=
f
(
t
,
x
(
t
)
)
\dot x (t) = f(t,x(t))
governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field
f
f
is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone
K
K
is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity.