The kinetics of homogeneous nucleation-growth processes under increasing supersaturation is investigated. The increase of the supersaturation is hereby caused by an appropriate variation of external parameters such as pressure and temperature. Analytic expressions are formulated for the dependence of the number of supercritical clusters both on the rate of change of the supersaturation and on time. In generalization of previous studies, both thermal and athermal nucleation are taken into consideration. It turns out that in dependence on the rate of change of the external parameters, either thermal (for moderate rates) or athermal (for higher rates) nucleation, dominates the process. It is shown further that, in the range where thermal nucleation dominates, the onset of nucleation-growth processes, i.e., the minimum value of the supersaturation required for intensive nucleation, depends weakly (logarithmically) on the rate of increase of the supersaturation. Criteria are formulated under which conditions the commonly employed assumption-independence of the nucleation-growth process on the way the initial unstable state is established-is applicable. As shown, quite generally these criteria are not fulfilled. In a further step of the analysis, simultaneously to external variations of the thermodynamic parameters, internally generated changes of the state of the system (depletion effects) are accounted for. For segregation processes in solutions (bubble formation), which are analyzed here as an example, such effects result from a decrease of the supersaturation due to the consumption of solute particles by the clusters of the newly formed phase. Basic characteristics of the nucleation-growth process, such as the maximum number of clusters formed in the system, are determined in dependence on both externally (rate of change of the external parameters) and internally (depletion effects) induced changes of the thermodynamic state of the system. It is shown, in particular, that the interplay of both factors is, in general, of comparable importance for the outcome of the nucleation-growth process.