Reedy model structures on oplax limits

You may be thinking of

Johnson, Mark W. On modified Reedy and modified projective model structures. Theory Appl. Categ. 24 (2010), No. 8, 179–208.

but his constructions (Definitions 3.3 and 5.2) have a fixed category at each object of $R$ and only the model structures are allowed to vary.

As a side note, a particular instance of this construction is used in these new notes about global homotopy theory, but here the indexing category is not even a Reedy category, but some sort of "enriched generalized Reedy category".