The language used to differentiate between different kinds of research involving more than one country is often very unclear. Terminology such as cross-country, cross-national, cross-societal, cross-cultural, as well as trans-national, trans-societal, trans-cultural is often used synonymously with comparative research, so what do you call research that transcends national boundaries but is not comparative?
Trying to find academics in the field to clarify terminology has led me to Kohn, Øyen, Hantrais and Mangen, Green and Hannerz ... so far ...
I am currently leaning towards Hannerz (1996) understanding. He classifies a study involving more than one nation as international when the nation itself is the object of study, anything else as transnational. As I am not studying nations per se but a concept within national institutions, my study can qualify as a transnational study.