We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. 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 and number distinct parts, all accurate to $o(1)$.