There is a new and excellent paper by Vincent Armentano, Paul Niehaus, and Tom Vogl on this topic.  Here is one paragraph (not from the abstract):

We also see no cases in which changes in transfers (from public and private sources) played a dominant role. Among households that exited poverty, the share of income they obtained from transfers either rose slightly or fell substantially. Among those that entered poverty, the share generally rose substantially or fell slightly. Overall, the data are consistent with progressive redistribution, but not with transfer income accounting directly for a major share of the income gains that moved households above the poverty line. In this sense, the households that left poverty did so largely on their own…

Transitions out of agriculture accounted for a limited role. They did not account for the largest share, let alone the majority, of transitions out of poverty in any country.

Keep reading this article on Marginal Revolution.