A similar question to this post, but now row sums are zeros. I don't think the suggeted approach in the comment works anymore.
Given $$\dot x = A(t)x(t)$$ where $A(t)$ is a Metzler matrix whose row sums are zeros at all time $t$, prove that the state transition matrix of the system is a left (or right) stochastic matrix for all time $t$ and $t_0$.
Just need some hints on where to start.
If $M(t)$ is the state transition matrix from time $0$ to time $t$, then $$x(t)=M(t)x(0) \quad (*) \,.$$ Pick $x(0)={\bf 1}$ (the all ones vector) to find that for this initial condition, the solution of $$\dot x = A(t)x(t)$$ is $x(t)={\bf 1}$ for all $t$, because $A{\bf 1}=0$ by assumption.Thus by $(*)$, we have ${\bf 1}=M(t){\bf 1}$, i.e., $M(t)$ is stochastic for all $t$.