The resulting product, fatty acyl carnitine is the transported across the inner mitochondrial membrane by a transporter that exchanges it for a carnitine molecule. In the matrix, the carnitine on the end of the FA's is replaced by coenyzyme A in a reaction catalyzed by CPI II, thereby restoring the activated FA in its original form. This consists of four reactions that repeated, results in the serial release of 2C units. See the animation link. This set of reactions resembles the succinate to OAA reactions with which you are familiar.
Eukaryotes have unique membrane bound organelles that can carry out specialized sets of reactions.
Peroxisomes are generally used to metabolized very long chained fatty acids and does so with distinct enzyme isoforms, but a very similar sequence. The first reaction is the most different. People with this disorder cannot metabolize FA's with a length of 6-12 carbons.