two approaches come to mind instantly. Firstly, from a sociological perspective, you might follow the figurational approach of Elias and Dunning. In that line of research, Joe Maguire has written on Global Sport (1999), and Dominic Malcom for instance has published on the development of Cricket in British colonies. The core argument is that western sport has been spreading in a process of stronger global interdependencies...
Secondly, if you are rather interested in state strategies etc than in people's behaviour, you might consider the growing sport investments of many countries as strategy to gain soft power. This concept from International Relations (Nye, 2004) is currently used a lot in sport policy research. For instance Paul Michael Brannagan has worked extensively on Qatar and its strategies concerning hosting sport mega events etc.
Dear Jan, thanks for your answer. Since my study has a cultural history perspective, I will definitely look into Maguire arguments. Different than other colonies, the Gulf British protectorates (Qatar, Bahrain and UAE) did not have a mission civilisatrice nor a religion missionary movement. Thus, the assimilation of western-sport does not follow (?) the 'traditional' civilised aspect and also could not (?) be consider a symbolic 'battle field'. Paul's studies about Qatar using Soft Power strategy are great, however do not apply in historical terms.
I made a research about how peruvian people asimilated football. I used Modernity At Large, from Arjun Appadurai, the chapter "Jugando con la modernidad: la descolonización del Cricket en India". Maybe this chapter can help you.
Indeed, Appadurai's paper is very interesting in relation to sport for national unity amid a process of decolonization-modernity (although India colonisation was quite different than Qatar).
The English version of the article is available online. Here is the full reference: Appadurai, A. (2015). Playing with modernity: the decolonization of Indian cricket. Altre Modernità, (14), 1-24.