In this paper, we determine the almost sure values of the $Φ$-dimensions of random measures supported on random Moran sets that satisfy a uniform separation condition. The $Φ$-dimensions are intermediate Assouad-like dimensions, the (quasi-)Assouad dimensions and $θ$-Assouad spectrum being special cases. Their values depend on the size of $Φ$, with one size coinciding with the Assouad dimension and the other coinciding with the quasi-Assouad dimension. We give many applications, including to equicontractive self-similar measures and $1$-variable random Moran measures such as Cantor-like measures with probabilities that are uniformly distributed. We can also deduce the $Φ$-dimensions of the underlying random sets.