We enrich the class of power-constructible functions, introduced in [CCRS23], to a class C M,F of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of C M,F. We furthermore identify a subclass C C,F of C M,F which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and rightcomposition with subanalytic maps. This class is also stable under parametric integration, under taking pointwise and L p-limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in C C,F. Contents