A random composition of n appears when the points of a random closed set ℛ̃⊂[0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K_n of this composition and other related functionals under the assumption that ℛ̃=ϕ(S_•), where (S_t,t≥0) is a subordinator and ϕ:[0,∞]→[0,1] is a diffeomorphism. We derive the asymptotics of K_n when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function ϕ(x)=1−e^−x, we establish a connection between the asymptotics of K_n and the exponential functional of the subordinator.