the lyman cyclone tumbler is rated for 1000 223 cases, or right around 13 lbs of brass.
so as long as you're obeying their suggested volume in the tumbler (no more than 3/4 full) and you stay under that weight you should be fine .
one thing to consider when mixing brass is that SOME types will nest in others. this is problematic for straight walled cases when mixed -so something to be aware of. this applies to dry tumbling too, but still worth noting.
my thumblers tumbler is rated for 12 lbs in the barrel, so with 5 lbs of pins, i use 2 lbs (1 quart) of water, and 5 lbs batches of brass.
get some lemi shine, forget whatever's in that OEM packet, and steal use some liquid dish soap off the kitchen sink. $5 of lemi shine and a $2 bottle of dawn will last you for *YEARS* of cleaning.
when you use lemi-shine, rinse very well in cold water when you're done, and with any wet cleaning method a good drying process will help reduce water spots.
I shake mine off first in the media seperator, then lightly towel dry to get the big drops off, and then put them in an old (yard sale special) food dehydrator to finish drying the insides of the cases thoroughly. this takes ~2 hrs.
probably one of the best $5 i've spent recently on reloading stuff when i found this dehydrator
HTH