We examine Dirichlet series which combine the data of a distance function, u, a homogeneous degree zero weighting function, and a multivariable Dirichlet series, K. By using an integral representation and Cauchy's residue formula, we show that under certain conditions on K, such functions extend to meromorphic functions on the complex plane, or to some region strictly larger than the domain of absolute convergence, and have real poles and polynomial growth in vertical strips. When the weighting function is identically 1, we also do this for u which come from completely nonvanishing polynomials on the positive orthant. Using standard Tauberian results, this allows us to deduce estimates for counting functions of points in expanding regions. We show that some of these results can be generalized to multivariable mixed zeta functions, and we use these to prove relations between coefficients of Laurent series of different Dirichlet series at s=0.