Unfortunately it is very rare to have a switch barrel rifle that maintains zero between barrels.
Mine are very close, but still different if using the same scope.
Another option, more expensive obviously, is to have seperate scopes zeroed in QD mounts for each barrel.
If using the same scope.
You need to sight in your most used barrel, zero your turrets, switch barrels and shoot without adjusting the scope and measure how far off you are, take notes once you have re-sighted THAT barrel.
Because you zeroed your turrets, you can then adjust back to zero when you switch back to the first barrel.
I have 2 rifles, Win Model 70's, with 3 barrels each, each barrel requires a different zero when using the same scope. I try to tune my loads for similar impact, but it doesn't always work. My 264WM barrel shoots 4" high and 3" right compared to my 270 Weatherby barrel, and my 300WM barrel is the one my turrets are zeroed with and the 270 Weatherby barrel shoots to same elevation but 2" right of the 300WM barrel.
My other rifle is similar, it has barrels in 22-250AI, 6.5x47 Lapua and 260AI.
I have 2 F-Class rifles that use different barrels depending on range based on Rem 700 actions, they both require very little adjustment for zero, as the barrel shoulders and the action mating services are precisely fitted to one another. I can spin on one, shoot a group, then spin on another and shoot a group with less than 1/2" difference in POI.
Cheers.