Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}_X(1)$, for any coherent sheaf $F$, we have $$H^i(X,F(m))=0,\quad m>>0$$ for every $i>0$ (i.e. there's an $m$ big enough such that the vanishing occurs for any bigger multiple $m'>m$ of the polarization and for every index $i$), where $F(m)$ denotes the twist $F\otimes\mathcal{O}_X(m)$.

If $D_*:=\{ D_m\}_{m\in\mathbb{N}}$ is a sequence of divisors on the projective variety $X$, we say that $D_*$ *satisfies SV on* $X$ if, for every coherent sheaf $F$ on $X$,
$$H^i(X,F(D_m))=0,\quad m>>0$$
for every $i>0$, where as usual $F(D_m)=F\otimes\mathcal{O}_X(D_m)$.

I'm mostly interested in the case in which:

($\star$) $X$ is a normal surface over $\mathbb{C}$.

Q1.Is there some "general" and "nontrivial" criterion for a sequence $D_*$ to satisfy SV? E.g. $D_*:=$ a subsequence of $\{m\cdot D_0\}$ for a fixed ample $D_0$ clearly satisfies it, but it's a trivial consequence of the usual SV.Also, is there some such property that is valid for every $X$? (For example, could it be something like: $D_*$ eventually forms a subsequence of a sequence of ample divisors which is "increasingly positive" in some specified sense?)

Narrowing down the question (and assuming ($\star$), if it is of any use):

Q2.Assume that $D_*$ is a sequence of (not necessarily effective)ampledivisors such that $D_{m+1}-D_m$ is effective. Does $D_*$ satisfy SV?