This thesis comes out at a time when the debate on between-state and social group demographic and health inequalities continues to be largely debated based on most recent available information. However, analyses based on the recent demographic and health data have serious limitations in terms of understanding the true trajectories of between-state and social group inequalities. With the changing demographic scenario and most demographic indicators progressing towards the final stages of transition, the major concern in India has been heterogeneity across region and socioeconomic groups. While the demographic heterogeneity in India is well known, how far the differences are converging or diverging has been a matter of great interest. This study attempted to develop a comprehensive framework to study the demographic transition, convergence and its linkage with health inequalities in India. It advances the empirical examination of demographic convergence assessment by assessing convergence not only in the averages, but also in the absolute and relative inequalities in population and health indicators. The analyses foster that while economic variables are diverging, fertility variables are converging in recent years. Perhaps, it re-affirms the fact that fertility transition is not critically related to economic factors in India. The mortality convergence has some setbacks in recent years. The child health care utilization has shown converging although at a lesser magnitude. Moreover, fertility decline is associated with increasing in child health inequalities in a context where there are pre-existing socioeconomic inequalities. Inequality is often a consequence of progress. Not everyone gets rich at the same time, and not everyone gets immediate access to the latest life-saving measures. Further, growth, inequality, and catch up are the both sides of the same coin. The dark side is what happens when the process is hijacked, so that catch-up never comes. Powerful and wealthy elites have choked off demographic and health progress before, and they can do so again if they are allowed to undermine the institutions on which broad-based demographic and health progress depends. Now, it is confirmed from this study that a mere reduction in the number of children or fertility convergence in averages are not helping in distributing the fruits of demographic progress equally among the children of different states and socioeconomic groups. How far children and their development (particularly in terms of health) are faring during the progress of demographic transition is an important concern for population policy in India.