So, yes, as you said, Mexico had their own stuff going on with them being invaded by the French. The French Emperor Napoleon III invaded the country due to unpaid debts among other things, and installed the Austrian Archduke Maximilian as Emperor of Mexico. Meanwhile, the deposed democratic government of Benito Juárez fled into the Mexican countryside to continue his own government in internal exile, quite literally while riding around the country in a stagecoach.

As far as I’m aware, because of the internal disputes, Mexico and the United States didn’t really have much major interaction during this time period. Abraham Lincoln and several members of his cabinet were sympathetic to Juárez. There may have been plans put forward by Lincoln to supply Juárez with funds and weapons in order to restore the democratic government. Peace negotiations between the Union and the Confederacy included talks about intervening against the Mexican imperial government following the conclusion of the Civil War. However, Lincoln‘s assassination brought an end to such plans.

To avoid possible repercussions, some Confederate soldiers fled abroad. Some may have gone to Mexico, but the majority of Confederate émigrés travelled to Brazil, where slavery was still a powerful institution until 1888.