This question can actually be tackled from many directions and many non-western societies can be examined. For the purpose of this exercise we will take a look at two non-western countries of that era, The Empire of Japan and The Russian Empire (which was technically a European power but not considered part of the Western World). It should also be noted that economic development, urbanization and class conflict all tie in together and were also responsible for the rise in the women’s rights movement. Let us just briefly examine this connection before we address the main theme of the question. Economic development led to mass urbanization as people from the countryside moved to the cities to serve the growing industries. This urbanization in turn led to class conflict across socio-economic lines as people competed for resources and power. The women’s rights movement was born out of these conflicts in order to carve a greater societal position and status for women. In regards to non-western societies the position of women’s suffrage plays a minimal role until the later parts of the 20th century (so there is no need to go into great detail on that).
The Empire of Japan experienced massive industrialization and economic growth in the two decades prior to the 1890’s which continued on to the 1900’s. In less than a generation they transformed themselves from a feudalist medieval society to an industrialized world power. The challenge for them was either to modernize or face conquest and colonization by European powers, like what happened to most of the rest of Asia. They succeeded in that challenge with some domestic opposition (from the traditional samurai class), which was at the end negligible and futile.
On the other side of the spectrum was The Russian Empire, they were also medieval and feudalist but failed to successfully implement necessary societal reforms. In the years between 1890 and 1914, The Russian Empire had numerous serious civil conflicts, internal uprising and clashes between the social classes. They had also in 1905 lost a major war to the Empire of Japan and had to agree to humiliating terms. The Russian Empire never made meaningful political reforms that appropriately satisfied or empowered the social masses. This error led to the rise of revolutionary elements and dissenters, whose agitations eventually led to the collapse of the empire.