For many pairs of degrees, the random lower-degree regular graph can be embedded inside the random higher-degree regular graph with high probability, and unions of random regular graphs can mimic a single random regular graph.
Den Hollander, Probability theory: The coupling method, 2012, Lecture notes available online (https://prob.math.leidenuniv.nl/lecturenotes/CouplingLectures.pdf)
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Monotonicity and decompositions of random regular graphs
For many pairs of degrees, the random lower-degree regular graph can be embedded inside the random higher-degree regular graph with high probability, and unions of random regular graphs can mimic a single random regular graph.