In recent times, considerable insight has been gained from comparing certain statistics of random matrix ensembles to the corresponding statistics taken over families of $L$-functions. We will look the transitions between ensembles of some such statistics, and for moments see what $L$-function-related conjectures naturally follow. Relating to an example family used above, an explicit formula will also be given for a shifted second moment taken over the family of $L$-functions of level $1$ cusp forms.