I've been investigating the history of Mali to try and determine how current events and their economic situation could be compared to previous struggles in regions nations. National sovereignty and stability/legitimacy of government is what I have been looking at. Was that was an issue during this period?
In general my understanding is that the former empire was split up among the former emperors offspring. I feel that my understanding of those separate kingdoms is uninformed and thought maybe there is a lack of historical information from that period. I know that Timbuktu had been a center of knowledge and culture for the region before becoming a part of the former empire. Did it carry on doing so after?
Effective Malian power ceased much earlier, around 1433, with the Tuareg conquest of Timbuktu. It passed to the Sonni Dynasty, headquartered in Gao, which formed the nucleus of the much larger tributary empire of Songhay. Neither was especially well centralized; each required a balance of benefit and coercion to remain under control but it was never absolute outside the Niger bend. Mali became a rump state on the upper Niger near Niani.
Songhay collapsed after defeat at the hands of the Moroccans at Tondibi in 1591. Part of the reason it had been rotting out was the succession issue, although Songhay was far more heavily bureaucratized and theoretically could have offset ineffective central leadership. The bigger problem was that the gold and other goods that had been going towards the north now went to the south, to the Europeans on the coast, who were bringing goods from Asia much more cheaply than they passed through the desert. The state was slowly going bankrupt--it couldn't afford infighting, or even the cavalry that maintained its hold across a vast network of tributaries.
So Songhay fractured into a number of smaller states, one run by the remnants of the Moroccan invaders (the Arma), one very abbreviated Songhay, and a clutch of other smaller states. The wealth necessary to attract, and then to hold together, a larger empire was no longer present, so political authority fragmented; you had kingdoms of Segu and Mesina, but they were smaller centralized states. That was largely the case until the era of the Jihads, after the late 1700s, which produced a new wave of states including Tukolor (the state of al-hajj Umar Tal after the 1840s) in part of that region, together with a rural Islamic majority that was a new development.
There was however no meaningful modern sense of national belonging or identity in Mali, or in the successor states; people identified by localities and social networks, not by government or ethnic label as we think of it. Nehemiah Levtzion covers some of this in work on Mali, maybe best the History of Islam in Africa. So yes, Timbuktu survived Mali, and even flourished after it, with people who identified as part of Timbuktu--but it began to decline as Songhay's wealth and patronage did.