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In this paper, we determine the almost sure values of the $\Phi$-dimensionsof random measures $\mu$ supported on random Moran sets in $\R^d$ that satisfya uniform separation condition. This paper generalizes earlier work done onrandom measures on homogeneous Moran sets \cite{HM} to the case of unequalscaling factors. The $\Phi$-dimensions are intermediate Assouad-like dimensionswith the (quasi-)Assouad dimensions and the $\theta$-Assouad spectrum beingspecial cases.  The almost sure value of $\dim_\Phi \mu$ exhibits a threshold phenomena, withone value for ``large'' $\Phi$ (with the quasi-Assouad dimension as an exampleof a ``large'' dimension) and another for ``small'' $\Phi$ (with the Assouaddimension as an example of a ``small'' dimension). We give many applications,including where the scaling factors are fixed and the probabilities areuniformly distributed. The almost sure $\Phi$ dimension of the underlyingrandom set is also a consequence of our results.

Assouad-like dimensions of a class of random Moran measures II -- non-homogeneous Moran sets