Since my answer is long, here is a loose heuristic summary answer: A permutation $\pi$ such that $|\pi(i+1) - \pi(i)|$ is large for many values of $i$ occurs less frequently than permutations where such values are small.

The first results on this might be from Bandt and Shiha's 2005 paper Order patterns in time series, which gives the probability that $\pi \in S_n$ occurs as an ordinal pattern for AR(1) processes and fractional Brownian motion for $n = 3,4$. It includes Douglas Zare's answer for the case of ordinary Brownian motion. Another $n = 3,4$ answer is given in DeFord and Moore's Random walk null models for time series, which gives the probability that $\pi \in S_n$ occurs in a walk with uniform steps on $[b-1,b] \; \; \left(\frac{1}{2} \leq b \leq 1\right)$ as a piecewise-defined polynomial in $b$.

In John Mangual's 2012 answer, there is a suggestion to "look for general properties" instead of directly computing the distribution on $S_n$. This happens in The frequency of pattern occurrence in random walks, where Elizalde and Martinez showed that certain pairs of permutations occur with the same probability *regardless of choice of density function for the steps*. A matrix $L_\pi$ is introduced that maps the positive orthant to the region of steps that generate $\pi$. They show that if $L_\pi$ can be obtained from $L_\tau$ by permuting its rows and columns, then $\pi$ and $\tau$ occur with the same probability regardless of choice of step density function. The converse is conjectured to hold. A characterization for the pairs of permutations with equal frequencies independent of density choice is given in Martinez's thesis Equivalences on patterns in random walks via the all-subset (resonance) hyperplane arrangement.

A generic integral that calculates the probability that $\pi \in S_{n+1}$ occurs is $\int_{\mathbb{R}_{>0}^n} g(L_\pi(\mathbf{x})) \mathrm{d} \mathbf{x}$, where $g$ is the joint density function for the steps. The preprint Ordinal pattern probabilities for symmetric random walks uses this to compute or compare probabilities for various density functions. In particular, an answer is given to the question about steps drawn from the uniform distribution on $[-1,1]$. For that choice of density, the probability that $\pi \in S_{n+1}$ occurs is $\frac{K_\pi}{2^n n!}$, where $K_\pi$ is the size of a weak order interval in the affine Weyl group. A complicated elementary description of $K_\pi$ is that it counts the number of strictly upper triangular matrices $M$ such that $M_{i,j} = 0$ whenever there exists $k$ such that $\pi(k) \leq i < j \leq \pi(k+1)$ or $\pi(k+1) \leq i < j \leq \pi(k)$ subject to the constraint, due to Shi, that $M_{it} + M_{tj} \leq M_{ij} \leq M_{it} + M_{tj} + 1$ whenever $i < t < j$. Code that calculates this distribution is given here.

This provides a description of the extremes: The identity and its reverse occur with probability $\frac{1}{2^n}$, while any permutation where $1$ and $n+1$ are consecutive occurs with probability $\frac{1}{2^n n!}$. A permutation where $|\pi(i) - \pi(i+1)| = m$ for some $i$ is guaranteed to have probability less than or equal to $\frac{1}{2^n m!}$.

For the mean zero Gaussian, we can compare permutations by looking at the matrix $\mathrm{lev}(\pi)$ whose $ij$-th entry counts the number of positions $k$ such that $\pi(k) \leq i,j < \pi(k+1)$ or $\pi(k+1) \leq i,j < \pi(k)$. If $\mathrm{lev}(\pi)_{ij} \leq \mathrm{lev}(\tau)_{ij}$ for all $i,j$, then $\pi$ occurs more frequently than $\tau$. (This is a somewhat precise version of the heuristic this answer opened with.) For exact probabilities, there is a decent answer when $\pi$ is a permutation such that $|\pi^{-1}(i+1) - \pi^{-1}(i)| \leq 2$, which applies to any symmetric density function. For such permutations, the probability is given by
$$\frac{1}{2^n \prod_{j=1}^n \mathrm{lev}(\pi)_j},$$
where $\mathrm{lev}(\pi)_j$ is the number of positions $k$ such that $\pi(k) \leq j < \pi(k+1)$ or $\pi(k+1) \leq j < \pi(k)$. This expression applies to every $\pi \in S_{n+1}$ when the steps are drawn from a mean zero double-exponential distribution.