  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.
