(Revised to assume $\text{support}(X_i)=\mathbb{N}$ in response to counterexamples noted in comments.)
Let $S_n=X_1+...+X_n$, where the $X_i$ are i.i.d. with $\text{support}(X_i)=\mathbb{N}$ and $0<VX_i<\infty.$ Then we know by the CLT that $$\lim_{n\to\infty}P({S_n<ES_n})={1\over 2}.$$
Q1: What are necessary and/or sufficient conditions for the convergence of $P({S_n<ES_n})$ to be monotonic?
For the $X_i$-distributions that I've examined numerically (e.g., Negative Binomial and Poisson), computations suggest that $EX_i\in\mathbb{N}$ is neccesary & sufficient -- but I've been unable to prove it.
For example, if $X_i\sim\text{Geometric($p$)}$, then $EX_i\in\mathbb{N}$ iff $1/p\in\mathbb{N}$, and indeed the convergence appears monotonic iff $p$ is the reciprocal of a positive integer.
Similarly, if $X_i\sim\text{Poisson($\lambda$)}$, then $EX_i\in\mathbb{N}$ iff $\lambda\in\mathbb{N},$ and indeed the convergence appears monotonic iff $\lambda$ is a positive integer.
Here are some typical plots of $P({S_n<ES_n})$ v. $n$:
$X_i\sim\text{Geometric($p=1/3$)},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \text{Geometric($p=1/3.1$)}$
$X_i\sim\text{Poisson($\lambda=1$)},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \text{Poisson($\lambda=1.1$)}$
Here's my attempt to prove (strict) monotonicity when $X_i\sim\text{Geometric($p=1/3$)}$; i.e., $S_n\sim\text{NegativeBinomial($n,p=1/3$)},$ for which $ES_n=nq/p=2n$, with $q=1-p.$ The pmf of $S_n$ is the following: $$P(S_n=k) = \binom{k+n-1}{n-1} p^nq^k\quad(k=0,1,2,...),$$ and we want to show that for all $n$, $$\begin{align}P(S_{n+1}<ES_{n+1}) &< P(S_{n}<ES_{n}) \\[2ex] \sum_{k=0}^{2(n+1)-1}\binom{k+n}{k}p^{n+1}q^k &< \sum_{k=0}^{2n-1}\binom{k+n-1}{k}p^{n}q^k\\[2ex] \sum_{k=0}^{2n+1}\binom{k+n}{k}q^k &< \sum_{k=0}^{2n-1}\binom{k+n}{k}{3n\over k+n}q^k\\[2ex] \sum_{k=2n}^{2n+1}\binom{k+n}{k}q^k &< \sum_{k=0}^{2n-1}\binom{k+n}{k}\left({3n\over k+n}-1\right)q^k\\[2ex] \binom{3n}{2n}q^{2n}+ \binom{3n+1}{2n+1}q^{2n+1}&< \sum_{k=0}^{2n-1}\binom{k+n}{k}\left({3n\over k+n}-1\right)q^k\\[2ex] {12n+5\over 12n+3}\binom{3n}{2n}\left({2\over 3}\right)^{2n}&< \sum_{k=0}^{2n}\binom{k+n}{k}\left({2n-k\over k+n}\right)\left({2\over 3}\right)^k\tag{*}\\[2ex] ...\ & ?\ ... \end{align}$$
Q2: Any suggestions on how to proceed, or alternative approaches? (I tried using the cdf in the form of a regularized beta function, but it led to similar difficulty.) Anyone know of binomial identities/inequalities that will further simplify (*)?