If you look at the data on the web for shot gun shells or in the manuals you will find that most combinations of components including different powders are listed. There is definite overlap of loads with different primers so sometimes a change to a different primer will still make a safe load but other times it may not so check the data each time you change any of the components.
A change in primer may make the shell slower or faster. Here is an example of loads from the data at Hodgdon.com for 12ga 1-1/8 oz in AA Winchester shells with WAA12 wads using Win Super Hcp powder and the affect of different primers. Each load is the maximum listed for the primer.
PRIMER WAD GRS PRESSURE VEL
Ched. 209 WAA12 20.5 11,400 PSI 1255
Fed. 209A WAA12 18.9 10,300 PSI 1200
Fio. 617 WAA12 22.5 11,400 PSI 1310
Rem. 209P WAA12 20.8 10,700 PSI 1255
Rio G-600 WAA12 20.8 10,600 PSI 1255
Win. 209 WAA12 20.9 11,300 PSI 1255