The Becker-Döring (BD) nucleation equation is known to predict a narrow double-exponential front (DEF) in the distribution of growing particles over sizes, which is due to early transient effects. When mass conservation is included, nucleation is eventually exhausted while independent growth is replaced by ripening. Despite the enormous difference in the associated time scales, and the resulting demand on numerics, within the generalized BD model the early DEF is shown to be crucial for the selection of the unique self-similar Lifshitz-Slyozov-Wagner asymptotic regime. Being preserved till the latest stages of growth, the DEF provides a universal part of the initial conditions for the ripening problem, regardless of the mass exchange mechanism between the nucleus and the matrix.