Let $\{x_i\in\mathbb{R}\}_{i=1}^{K}$ be a set of random univariate variables, each of which is distributed normally as $x_i\sim\mathcal{N}(\mu_{x,i},\sigma_{x,i}^2)$. Note that $x_i$'s are pairwise independent.
Also, let $\{y_i\in\mathbb{R}\}_{i=1}^{K}$ be the set of random variables, where each variable $y_i$ is given as the following non-linear transformation of $x_i$: $$ y_i=\frac{e^{x_i}}{\sum_{j=1}^{K}{e^{x_j}}}. $$ How could we find $y_i$'s mean $\mu_{y,i}$ and variance $\sigma_{y,i}^2$?
You can use Unscented Transformation to find mean and covariance. Let $y = f(x)$ be a linear or nonlinear mapping from $x$ to $y$. Given that you know the mean and covariance of $x$, you can find mean and covariance of $y$ by generating sigma points corresponding to $x$.