In this paper, the authors introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) that are analytic in a half-plane. Then they study composition operators on these spaces and provide criteria for boundedness and compactness when the symbol inducing a composition operator is an affine function. They also study the cyclicity property and give a characterization for the direct sum of the identity and a weighted forward shift operator on $l^2$ to be cyclic.