  We define the notion of asymptotically free for locally restricted
compositions, which means roughly that large parts can often be replaced by any
larger parts. Two well-known examples are Carlitz and alternating compositions.
We show that large parts have asymptotically geometric distributions. This
leads to asymptotically independent Poisson variables for numbers of various
large parts. Based on this we obtain asymptotic formulas for the probability of
being gap free and for the expected values of the largest part, number of
distinct parts and number of parts of multiplicity k, all accurate to o(1).
