X-Rite ColorChecker Passport and non-Adobe software
While the X-Rite ColorChecker Passport comes with software, the software only works with Adobe RAW converters (ACR, Adobe Camera Raw) to create a custom color profile.
So what do you do if you want to use another manufacturer's RAW conversion software?
The solution lies in knowing what each color square is "supposed to be", so that you can revert the taken image of the Passport color target back to the standard colors X-Rite designed for the system.
Fortunately, X-Rite and a couple of others have provided most of what we need:
X-Rite Colorimetric values for ColorChecker Family of Targets, this is the official sRGB values for each of the color squares, plus it provides Lab values as well.
sRGB values of X-Rite color checker passport, echoes these values, but also superimposes the values onto an image of the color squares of the Passport target, to make it a little easier to use.
But what if you want to use Adobe RGB color space, which can be valuable for landscapes, etc.? (I don't recommend Adobe RGB for skin tones because it allows for fewer red values than sRGB, and human skin can look better in sRGB.)
Fortunately, Robin D. Meyers provides another table of values for the Passport in Adobe RGB color space and also ProPhoto RGB:
ColorChecker Passport Technical Review, starting on page 7
Armed with the above information, open the starting RAW image of each session, the image which includes the Passport color chart, and perform basic white balance against the color chips designed for that use. Now you can use an eyedropper sampler in your software to check each color patch against the desired values from the charts above, and use your RAW color correction to make changes back to the standard values.
You don't necessarily have to check each color chip, only those critical to your "vision" of the scene.
Once you have made all the desired color changes, save the changes as a camera-color-correction profile and apply the profile to all appropriate images in the session set.
After this, apply any subsequent image processing you desire in your processing workflow.