I changed the subject line so as to not confuse folks that we are talking about people here and not the unit 6th MO Cav.

1. Yes men in Missouri often fought with different units. On the Union side they often were in several different Militia units raised for short periods of time. On the "Southern" side (note I did not say Confederate) many men served in the Missouri State Guard before being enlisted in to regular Confederate CSA units. There are also men who show up as "deserting" but just went home to be irregular guerillas in their home neighborhood. This is particularly true after the remnants of the MSG was sent east of the Mississippi River as CSA units in 1862. Once back home, they were recruited by any number of Southern leaning recruiters some acting with authority under the Partisan Ranger Act and some not, to join units recognized by the CSA or just acted as irregular fighters. So it's not unusual to find folks with ties to as many as 3 or 4 units. To add to this, many of the irregular units would winter and R&R in Texas, Indian Territory, and Arkansas and sometimes if wounded would then join a unit from those areas. And to top it off, we have the phenomenon of "drafting at the end of the gun barrel". This process basically occurred when units, Union or Confederate, needed men and whoever was stronger in a particular community or area would literally say "come with us or else". So there is more than a few that have both Confederate and Union service records!.

2. Exchanged POW's often were assigned to new units. By definition of exchange it was legal to do so by both armies. Parole on the other hand was supposed to keep folks out of the war for good and was generally associated with a bond being posted typically $1-2,000. Sometimes as much as $10,000. Real money in the 1860's. With that said, many paroled southern fighters re-enlisted. A large number did so in 1864 during Price's Raid.

Please feel free to post your ancestor's names and if you have a unit they were with and/or a DOB or place of residence there are many who can help open the door for you. There are many online sources to look at and I will try to fill in the source and link back so you can dig on your own. 6th Missouri Cav is an example, Edwards book is worth reading and is available online for free.
Further don't forget to use the search function on this bulletin board for example this thread from 2007 may be of interest:
http://www.history-sites.com/cgi-bin/bbs62x/mocwmb/webbbs_config.pl?md=read;id=8483 (Copy and paste into your browser)
Enjoy!