Regional convergence theory predicts that poorer places may catch up with richer ones as productivity and human capital accumulate, yet convergence remains uneven in many developing countries. Peru experienced substantial reductions in poverty between 2007 and 2017, but many districts that started among the poorest remained there a decade later. This paper examines which initial district characteristics are associated with upward movement in the poverty distribution. We find that the composition of local human capital is more informative than municipal fiscal indicators. Higher secondary educational attainment is strongly associated with upward mobility, while tertiary attainment exhibits a nonlinear relationship with persistent poverty among initially poor districts. We also find that secondary education in neighboring districts is associated with favorable re-ranking, although evidence on pre-period migration indicates that this relationship reflects both local complementarities and skill-selective sorting rather than pure spillovers. These findings highlight the importance of territorial human capital in shaping local convergence despite broad national poverty reduction.