That would be my guess - that the movement of the stars between shots was enough to bring lens distortion into the equation. Which lens are you using? If it's just a simple barrel/pincushion distortion then you might be able to fix it before stacking. But anything more complicated like mustache (in case you're using the Samyang/Rokinon 14 mm) would probably lead to issues.
Even with simple barrel distortion you might run into problems. The fixes are only approximate, intended to remove visible distortion in an image when viewed as a whole. But this use is a particularly extreme test of how perfect the correction can be, I wouldn't be fully surprised if there are still sections that are out of alignment. It's an interesting problem, I'm curious to hear how it plays out. That said, I'm sure you're far from the first to do this, so perhaps visiting an astro-photographer forum would be helpful.
Personally, I'd have Lightroom apply the lens profile correction, then export all the images to Photoshop, using the same post processing. If you're using the Samyang there's a profile you can download, but it only gets it close. You may have to do a bit of manual masking on specific areas to make sure everything is as sharp as you want it.