Suppose $f,g. u \in C^{1}((0,T) \times \mathbb{R})$ with$$ \partial_{t}(f+g) + \partial_{x}((f+g)u) = 0, \quad \text{on } (0,T) \times \mathbb{R}$$ where for each $t \in (0,T)$ we have $\text{spt}(f(t,\cdot)) \cap \text{spt}(g(t,\cdot)) = \emptyset$. Then do we have $$\partial_{t}f + \partial_{x}(fu) = \partial_{t}g + \partial_{x}(gu)= 0 \quad \text{on } (0,T) \times \mathbb{R}?$$
I think it should be true almost trivially but I don't know if I am making a mistake. Since $f+g$ is either equal to $f$ or equal to $g$ (Depending on which support you are in) then this should lead to the desired conclusion. Or is there some sort of pathological example I am missing?