Connections between colored restricted $b$-ary and ordinary partitions

We establish various connections between classes of colored and bounded ordinary partitions on one hand, and colored but not necessarily bounded binary and $b$-ary partitions on the other hand. Many of these results are based on special recurrence relations, some of which are new. We also obtain several classes of identities for sequences of colored $b$-ary partitions, and there are a few results concerning compositions.