What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$? Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$  and consider induced gradient-flow
$$
\dot X(t) = -\nabla L(X(t)).
$$

Question. Is there always a nontrivial scalar quantity (some kind of energy, maybe) which is conserved (i.e stays constant over time) under such a gradient-flow ?

Examples
As an example, let $A$ and $B$ be fixed matrices of size $n \times m$ and $n \times k$ respectively. For $U \in \mathbb R^{m \times m'}$ and $V \in \mathbb R^{m' \times k}$, define $L(U,V) := (1/2)\|AUV-B\|^2$. Here, the pair of matrices $(U,V)$ plays the role of the variable $x$ in the general scheme above (via trace inner-products, one can naturally give such pairs the structure of a euclidean space of dimension $mk(m')^2$). A simple computation reveals that the gradient of $L$ at $(U,V)$ is $(WV^T,U^TW) \in \mathbb R^{m \times k + m' \times k}$, where $W=W(U,V) := A^T(AUV-V) \in \mathbb R^{m \times k}$. Thus the induced gradient-flow is
$$
\begin{split}
\dot U(t) &= -W(t)V(t)^T,\\
\dot V(t) &= -U(t)^TW(t).
\end{split}
\tag{1}
$$
Now, for every $z \in \mathbb R^{m'}$, one may compute
$$
\begin{split}
&\frac{1}{2}\frac{d}{dt}\|U(t)z\|^2  - \frac{1}{2}\frac{d}{dt}\|zV(t)^\top\|^2\\
&\quad = \langle zz^\top U(t)^\top,-W(t)V(t)^\top\rangle - \langle zz^\top V(t),-U(t)^\top W(t)\rangle = 0,
\end{split}
$$
where the last equality is via cyclic permutation-invariance of trace of matrix products. Thus, for every $ z \in\mathbb R^{m'}$ the quantity $E_z(t) := \|U(t)z\|^2  - \|V(t)^\top z\|^2$ is conserved under the gradient-flow (1). In particular, taking $z$ to be on of the standard basis vectors of $\mathbb R^{m'}$ and summing over all these vectors, we have that $E(t) := \|U(t)\|_F^2  -\|V(t)\|_F^2$ is conserved.
 A: It turns out that Noether's Theorem  (about conservation from symmetry) gives an efficient machine for producing conservation laws for gradient-flow!

Indeed, consider once again a gradient-flow  $\dot x(t) = -\nabla \phi(x(t))$, where $\phi:\mathbb R^N \to \mathbb R$ is a sufficiently regular scalar-field. By a conserved quantity, I mean a functional $E:\mathbb R^N \to \mathbb R^M$ such that $E_c(x(t)) = E_c(x(0))$ for all $t \in \mathbb R$.

Symmetry Assumption. Suppose $G$  Lie-group of transformations $\mathbb R^N \to \mathbb R^N$ which leave $\phi$ invariant, i.e $\phi(g(x)) = \phi(x)$ for all $g \in G$ and $x \in \mathbb R^N$.

Let $c$ be an infinitesimal generator of $G$.
Then, it is shown in this paper Noether: The More Things Change, the More Stay the Same (see eqn 17 therein) that the quadratic form $E_c:\mathbb R^N \to \mathbb R$ defined by
$$
E_c(x) = \langle x, c(x)\rangle, \label{*}\tag{*}
$$
is conserved under the flow, i.e $E(x(t)) = E(x(0))$ for all $t \in \mathbb R$.
As a sanity check, let us recover the computation done in the OP, where

*

*$N=mm' + m'k$,

*$x=(U, V) \in \Theta:=\mathbb R^{m \times m'} \times \mathbb R^{m' \times k}$, where $\Theta$ is equipped with inner-product
$$
\langle (U, V), (R,S)\rangle := \langle U,R\rangle_F + \langle V,S\rangle_F,
$$

*$\phi(x) = U(U,V) := \|B-AUV\|_F^2$.

It is clear that $U$ is invariant to the following group of transformations $G := \{g_\lambda \mid \lambda \in (0,\infty)\}$, where each $g_\lambda$ is defined by
$$
g_\lambda(U, V) := (\lambda U, \lambda^{-1} V).
$$
This is a one-parameter Lie-group with infinitesimal generator $\epsilon(g)$ at $g \in G$ given by the equation
$$
g = \lim_{\eta \to \infty}(1+(1/\eta) \epsilon(g))^\eta = e^{\epsilon(g)},
$$
Solving this equation gives $\epsilon(g) = \log(g)\cdot \zeta$, where $\zeta \in G$ is defined by $\zeta(U,V) := (U,-V)$. Thus, an infinitesimal generator of $G$ is $\zeta$. We deduce from \eqref{*} that the conserved quantity is the quadratic form $E_\zeta(x) = \langle x, \zeta(x)\rangle$, i.e the quantity,
$$
\label{1}\tag{1}
E_\zeta(U, V) := \langle (U,V), (U, -V)\rangle_F = \|U\|^2_F - \|V\|_F^2.
$$

Actually, by considering a larger symmetry group $G := \{g_\Lambda \mid \Lambda \in \mathrm{GL}(m',\mathbb R)\}$ with $g_\Lambda(U,V) := (\Lambda U,V\Lambda^{-1})$, whose infinitesimal generators are easily seen to be $$\mathrm{LieAlg}(G):=\mathrm{gl}(m',\mathbb R)\zeta = \{\Lambda \zeta \mid \Lambda \in \mathrm{gl}(m',\mathbb R)\},\text{ with } \mathrm{gl}(m',\mathbb R)   = \mathbb R^{m' \times m'},$$ we can easily check that  for all $\Lambda \in \mathrm{LieAlg}(G)$, the real-valued functional $\widetilde E_{\Lambda\zeta}(U,V) = \langle (U,V),(\Lambda U,-\Lambda V)\rangle_F = \mbox{trace}(\widetilde E(U,V),\Lambda )$ is conserved, where $\widetilde E:\mathbb R^N \to \mathbb R^{m' \times m'}$ is the matrix-valued functional defined by
$$
\widetilde E(U,V) := U^\top U - V^\top V.
\label{2}\tag{2}
$$
The previous statement is equivalently to the statement that $\widetilde E$ is conserved. Conservation of $\widetilde E$ of course implies in particular, that the real-valued functional $E$ defined in \eqref{1} is conserved. To see this, simply note that $E(U,V)=\|U\|_F^2 - \|V\|_F^2$ equals the trace of $\widetilde E(U,V)$.
A: No.
For example, suppose $x_0 \in \mathbb{R}^N$ is an asymptotically stable equilibrium point of the gradient flow; suppose $- \nabla L (x_0)  = 0 $ and the matrix $- \nabla^2 L (x_0)  $ has $N$ negative eigenvalues. Then there exists an open set $U$ about $x_0$ such that every point $y \in U$ limits to $x_0$ in forward time. If there is some continuous conserved quantity $F : \mathbb{R}^N \to \mathbb{R}$, then $F(y) = F(x_0)$ for all $ y \in U$. That is, $F$ must be constant on the open set $U$. If $F$ is a polynomial function or an analytic function, it would need to be constant on all of $ \mathbb{R}^N$.
A similar argument can be made if instead the matrix $- \nabla^2 L (x_0)  $ has $N$ positive eigenvalues. There, an open set of points would converge to $x_0$ in backwards time, and so again any conserved quantity would need to be constant on that open set.
A: In the way you ask the question, I would say "yes" provided $L$ is $C^2$ or you otherwise know that an existence-and-uniqueness result applies to the solutions of the dynamics. In that case, let $t_{\min}(X_0)$ and $t_{\max}(X_0)$ be the minimal and maximal time of the maximal solution to the IVP
$$\dot{X}(t)=-(\nabla L)(X(t))\qquad X(0)=X_0\in \mathbb{R}^N.$$
The set
$$\Xi:=\{\{X(t)\in \mathbb{R}^N:\,X(0)=X_0,\,t\in (t_{\min}(X_0),t_{\max}(X_0))\}:X_0\in \mathbb{R}^N\}$$
of "images of maximal solutions" is a partition of $\mathbb{R}^N$. The function
$$f:\mathbb{R}^N \to \Xi: x \mapsto O\in \Xi \text{ where }x\in O$$
is conserved under the dynamics. Any left-composition of this function with another function yields another conserved function.
Gradient-flow structure is irrelevant for this to hold.
