Simple random walk (and non-backtracking walk) on random regular graphs exhibit the cutoff phenomenon [1]. The extension to graphs with degree sequences came later; see [2] for nonbacktracking walks and [3] for simple random walk where backtrackings cause additional difficulties.

In another direction, component structure at criticality was described in a well known paper by Aldous [4]. It was adapted to random regular graphs in [5], and extended to other degree sequences in [6] and [7].

References:

[1] Lubetzky and A. Sly,
Cutoff phenomena for random walks on random regular graphs. Abstract
Duke Mathematical Journal 153 (2010), no. 3, 475–510.

[2] Ben-Hamou, Anna, and Justin Salez. "Cutoff for nonbacktracking random walks on sparse random graphs." The Annals of Probability 45, no. 3 (2017): 1752-1770.

[3] N. Berestycki, E. Lubetzky, Y. Peres and A. Sly,
Random walks on the random graph.
Annals of Probability 46 (2018), no. 1, 456–490.

[4] Aldous D. (1997), Brownian excursions, critical random graphs and the
multiplicative coalescent. Ann. Probab. 25, 812–854.

[5] Nachmias, Asaf, and Yuval Peres. "Critical percolation on random regular graphs." Random Structures & Algorithms 36.2 (2010): 111-148.

[6] Bhamidi, Shankar, Remco Van Der Hofstad, and Johan van Leeuwaarden. "Scaling limits for critical inhomogeneous random graphs with finite third moments." Electronic Journal of Probability 15 (2010): 1682-1702.

[7] Riordan, O. "The phase transition in the configuration model." Combinatorics, Probability and Computing 21 (2012), 265--299 21, no. 1-2 (2011).