Monotonicity properties of the gamma family of distributions

For real $a>0$, let $X_a$ denote a random variable with the gamma distribution with parameters $a$ and $1$. Then $\mathsf{P}(X_a-a>c)$ is increasing in $a$ for each real $c\ge 0$; non-increasing in $a$ for each real $c\le -1\mathrm{/}3$; and non-monotonic in $a$ for each $c\in (-1\mathrm{/}3,0)$. This extends and/or refines certain previously established results.