In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count cyclically restricted and circular objects. This either fills gaps in the current literature by e.g. considering particular new patterns, or involves general progress, notably with locally restricted compositions over a finite group. We associate these compositions to walks on a covering graph whose structure is exploited to simplify asymptotic expressions. Specifically, we show that under certain conditions the number of locally restricted compositions of a group element is asymptotically independent of the group element. For some problems our results extend to the case of a positive number of subword pattern occurrences (instead of zero for pattern avoidance) or convergence in distribution of the normalized number of occurrences. We typically apply the more general propositions to concrete examples such as the familiar Carlitz compositions or simple subword patterns.