There is a "known true version", but it is basically what you have already as the converse.

**Defn** Let $(M,g)$ and $(N,h)$ be Riemannian manifolds, and let $\phi: M\to N$ be a smooth mapping. The *tension field* of $\phi$ is defined to be
$$ \tau(\phi) = \mathrm{trace}_g D \mathrm{d}\phi $$
where $D$ is the pull-back of the Levi-Civita connection on $N$ to $M$ via $\phi$.

**Defn** A mapping $\phi:M\to N$ is said to be "harmonic" if $\tau(\phi) \equiv 0$.

**Rmk** In the case $M$ and $N$ are Euclidean spaces, this reduces to the components of mapping being harmonic functions.

**Defn** Given a Riemannian manifold $(M,g)$, a vector field is said to be 1-harmonic-Killing if, denoting by $\varphi_t$ its corresponding one parameter family of diffeomorphisms, that
$$ \frac{\mathrm{d}}{\mathrm{d}t} \tau(\varphi_t) \Big|_{t = 0} = 0 $$

**Rmk** $\tau(\varphi_0) = \tau(\mathrm{id}) = 0$. So the definition of 1-harmonic-Killing asks that the flow to be linearly harmonic.

**Thm** A vector field $V$ is 1-harmonic-Killing if and only if
$$ \triangle X^\flat = \mathrm{Ric}(X,\cdot) \tag{1}$$
where $\triangle$ is the Hodge-Laplacian and $\mathrm{Ric}$ is the Ricci tensor.

**Rmk** In the Euclidean case, Ricci curvature vanishes, and (1) reduces to what you state as "harmonic vector field".

This result can be found in: Dodson, Trinidad Pérez, and Vázquez-Abal; Stepanov and Shandra; and Nouhaud. The first reference also contains examples showing that there exists 1-harmonic-Killing vector fields whose flow is not always harmonic.

Notice however, Remark 3.1 and Proposition 3.1 in the first-cited paper above, which combine to say that *on a compact Riemannian manifold with non-positive Ricci curvature, a vector field $V$ is 1-harmonic-Killing if and only if $V$ is a parallel vector field*. In particular, this implies that on such manifolds any 1-harmonic-Killing vector field will give rise to a harmonic flow. While this statement gives a positive answer to your question, we should note that in this setting $V$ is also necessarily Killing and hence the flow generated is a one parameter family of isometries, and hence the result can also be viewed as a rigidity statement.

A geometrically interesting fact is this:

Let $V$ be a vector field and $\varphi_t$ its local one parameter family of isometries. In the above we asked about the property of $\varphi_t$ being harmonic. If we replace the word harmonic by

- isometric; we get Killing vector fields;
- conformal isometric: we get conformal Killing vector fields;
- totally geodesic: we get affine Killing vector fields.

In all three of the above situations, having the "1-" version hold (meaning that the condition holds for $\varphi_t$ to first order as $t \to 0$) implies that the condition in fact holds (to infinite order). So in the literature we do not distinguish between 1-Killing vector fields and Killing vector fields.

One reason behind this distinction is the fact that *compositions of harmonic maps need not be harmonic*, in contrast to the other three cases mentioned above.

Anydiffeomorphism $\phi_0$ can be extended to a flow along a vector field $V$. $\endgroup$