Here is a new paper by Dev Patel Justin Sandefur, and Arvind Subramanian:

The central fact that has motivated the empirics of economic growthnamely unconditional divergenceis no longer true and has not been so for decades. Across a range of data sources, poorer countries have in fact been catching up with richer ones, albeit slowly, since the mid-1990s. This new era of convergence does not stem primarily from growth moderation in the rich world but rather from accelerating growth in the developing world, which has simultaneously become remarkably less volatile and more persistent. Debates about a middle-income trap also appear anachronistic: middleincome countries have exhibited higher growth rates than all others since the mid-1980s.

Here is the entire paper.  My general conclusion is that no particular model of convergence, or lack thereof, is correct, and it simply all depends on the historical period.