Is there any health convergence across countries?. You'll find the answer using the Global Burden of Disease study in a recent Lancet article (a must read).
Part of the answer depends on how the goals are framed—for example, what does convergence mean? In the development literature on economic convergence, convergence has been framed in terms of poverty rates or in terms of income inequality measured by the Gini coefficient or other measures of inequality. Work on convergence in life expectancy has tended to focus on measures of absolute difference rather than relative difference. We found unequivocal divergence in mortality rates for women aged 25–39 years and older than 80 years and for men aged 20–44 years and 65 years and older, similar to previous estimates of divergence of life expectancy at birth since the 1980s. In these age groups, both the Gini coefficient and the mean absolute diff erence in death rates are rising. In all other age groups, except girls aged 10–14 years, relative inequality is increasing but the absolute gap is narrowing.
For most countries, the general pattern of reductions in age-sex specific mortality has been associated with a progressive shift towards a larger share of the remaining deaths caused by non-communicable disease and injuries. Assessing epidemiological convergence across countries depends on whether an absolute or relative measure of inequality is used. Nevertheless, age-standardised death rates for seven substantial causes are increasing, suggesting the potential for reversals in some countries.If longevity is mostly improving everywhere, now the key issue should be to analyse global convergence on quality of additional years of life.
PS. Check p.3499 of the supplementary information. And p. 150 for changes in total Life Expectancy 1990-2013, (5,6 y men, 3,9 y women). If you read my previous post, you'll find interesting differences about what is going on with healthy life years for women.