There are actually a number of connections that have been proposed by historians, and debated quite widely. You might particularly be interested in Roger Anstey's The Atlantic Slave Trade and British Abolition. Some of them revolve around politics -- the rising British bourgeoisie (Whigs) were opposed to aristocrats (Tories) who had a great deal of money tied up in slave plantations, and therefore abolition was a way to undercut their political opponents. Others are more economic, suggesting that the middle class connected free labor to other freedoms necessary for economic mobility. Still other scholars seem to suggest that there arguments are too instrumental/materialist and deterministic, and that we should be focusing instead on real cultural shifts that are not really the result of such material (or even socio-economic) changes but rather entangled with them in more complex ways. I'm not sure personally which are correct, but I follow the debate with some interest.

Related question: were there any major successful preindustrial civilizations that did not use slavery? What about other types of forced labor like serfdom and indentured servitude?