The dynamics of family livestock farming in extensive livestock farming territories are crucial for the future of these territories. Indeed livestock farming is an activity anchored in a society, with its agro-food chains and its local environment. It provides products and multiple services: social and economic dynamics, desirable landscape, biodiversity… Family livestock farming has to reshape itself, and face up to local factors of change, such as demographic evolution or land pressure, and global factors like climate or market. The purpose of our communication is to shed light on the main processes of transformation of family livestock farming, based on a comparative analysis carried out on 8 extensive livestock farming territories : 2 in South America, 1 in Sub-Saharan Africa, 2 in the Mediterranean area and 3 in the French mountains. We propose a cross-reading of these processes based on three complementary considerations: i) the link between family and livestock farming, ii) the link of the farm to space and finally iii) the link of the livestock farm to agro-food chains. This reading informs us on the diversity of adaptation dynamics of family livestock farms. In all the territories we can demonstrate the influence of two driving forces for change on the construction of this diversity of family livestock farm adaptation dynamics. These driving forces are globalisation and territorialisation. We propose a stylised representation of this and discuss the tensions and/or complementarities which this diversity of dynamics creates within the territories.