The way the slave trade unfolded has parallels with some of the political and moral phenomena of today, especially regarding animal rights and exploitation.
One consistent phenomenon is that participants in the market reacts to incentives, and morals are of little importance in a permissive political environment. This is true for both individuals and states. In the case of the latter, African oligarchs did not oppose the expansion of the industry, but rather actively exacerbated it by engaging in conflict to capture more people to sell to Europeans. The incentive in this case was the goods coming from European traders in the form of guns and ammunition and, to go one level deeper, the warring within Africa that created the need for these products. In the case of individuals, the rise of kidnapping and raids with the motivation to sell people to slavers demonstrates the superiority of incentives over morals to the influence of behaviour.
The buying of slaves is also testament to the power of incentive. With the need for more labor to capitalise on the booming sugar industry, it was more convenient for elites in the Americas to bring in more slaves than to change their system of production to avoid causing harm to more human beings, not to say that this was something on their minds. Africans also may have stopped subjecting people to slavery when Britain demanded legitimate commerce, but this was not a strong enough incentive to incite such a change. Whether as a precautionary or capitalistic utility, an object of incentive goes a long way in deciding, hence predicting the moral behaviour of people.
The second lesson is that the ethical sacrifice, in the favour of a voiceless minority, of a profitable but nefarious system more likely and more readily occurs in a politically inclusive environment. Britain abolished slavery owing to the efforts of a small but highly motivated fraction of its population. The leaders of this movement were not themselves affected by slavery, but rather has altruistic motives for abolishing it. In a system where only few have any influence on laws, it would be highly unlikely that such people would somehow happen to be on power. The fact that ordinary people had some political power also means that slaves, much more relatable to them than to aristocrats, stood a better chance of having allies within the privileged population.
In contrast to, and complimenting the point made by, the change that occured in Britain, internal African slavery did not dissipate as easily. This even after the drop in external demand for slaves and slave-produced goods. Lack of political centralisation, and even where it existed, the exclusivity of political power to oligarchs, strongly defied the abolition of local slave use. It would have suited those in power for the economic system that was already benefiting them to remain the same, and where there was weak or no administrative power, it would be almost impossible for oppressors to suddenly have a change of heart and stop kidnapping and enslaving people. Hence Africans continued to enslave each other even upto the 20th century.
Another reason for the disparity between African and European abolition of slavery could be the fact that Africa had an economy that was heavily dependent on slaves. It certainly helped Britain that they had a parallel free market economy that would expand to fill the gaps left by the ban on slavery. It is also likely that the greater efficacy of free workers compared to slaves, as they were juxtaposed, made the abolition of slavery seem more like a sensible change. This free market economy is underpinned by the presence of more pluralistic politics.
This ties in with modern day animal welfare because industries that exploit animals are an important part of the economy, yet the conditions required to ban them are already met in some countries. One more lesson is that change, within these conditions, can come through the efforts of a small but persistant group pressuring the government and these industries. It will be difficult, but it can be done.