Empirical evidence in the sparse literature on poverty convergence currently relies on cross-sectional analysis, where Less Developed Countries (LDCs) starting out poorer are found to have enjoyed no faster subsequent poverty reduction during the past three decades than those starting out richer, as initial poverty retards growth and makes it less effective in reducing poverty. Applying a dynamic panel approach to four panel data sets including two covering some 90 LDCs from 1977-2014 and two covering, respectively, 42 Ethiopian and 33 Rwandan regions during 1995-2010 and 2000-2010; this study finds that LDCs as well as Ethiopian and Rwanda regions do enjoy faster subsequent poverty reduction within themselves as their initial poverty incidences decline over time. Our analysis also finds initial poverty as having little direct effect on growth in LDCs, except in SSA where higher initial poverty is associated with faster growth.