Here are two examples, both from articles in Organization Studies, of what some linguists call the new meaning of the phrase "this begs the question". Others call it the wrong meaning.
If, as Adler argues, paleo-Marxism has been ‘eclipsed by neo-Marxism, of which LPT is an exemplar’, this begs the question: which labour process theorists have taken this retrograde step resulting in the inadequate acknowledgement of socialization?
In the former conception, ‘even the question of “internal”organization and administration now becomes related to an outside network of relative prices and costs’ (Robbins 1984: 71), which really begs the question of how much organization is left in this conception of ‘internal’ organization.
Originally, one would say "this begs the question" only to indicate a petitio principii, i.e., the act of assuming an answer to the very question you are addressing. Using it in this way, one would not follow "this begs the question" with a specific question (as in the two examples here). Instead, one would account for the sense in which an argument assumes precisely what it is supposed to prove. Today, however, many writers use it to mean simply "this raises the question".
Whenever that is the intended sense in a text I am editing, I normally change it accordingly, replacing "begs" with "raises". I do this for two reasons. First, it captures the intended meaning just as well; second, it avoids the unnecessary criticism of the reader who insists on the technical sense of "begs of the question".