Can photoshop cc stack and align photos produced from lenses with barrel distortion

I have a series of astrophotos that include terrestrial foreground and the milky way in the background.  The images were taken with a wide angle lens at f/2.8.  To increase the signal to noise ratio I would like to stack the images; however, due to the rotation of the earth the position of ht milky way relative to the foreground changes in each shot.  To correct for this I have imported the raw images as layers from lightroom and made a copy of all the images and transformed one copy of all the images into a smart object for stacking the foreground.  On the background (sky) copy, I made a layer mask that masked out the foreground for each photo.  Next I used the edit > auto-align-auto feature to align the images.  Next I deleted the layer masks and stacked the images using layers > smart-object > stack mode > median and photoshop did stack the images.  While the center and center top of the images did stack with a good image and an increase in the SNR, the rest of the photo came out blurred/smeared.  I have tried this many times and I keep getting the same thing.  Is barrel distortion causing this and if so is there anything I can do to correct for it?  I am attaching a before and after photo here, so hopefully you can see what I am talking about.  It is only aligning part of the sky and is just smearing the rest.

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.
Edit: I could see it just being a PS alignment issue.  There's a lot of stars there.  I'm not sure how the algorithm works, but I'd assume it just lines up a few points.  If it messed up one of the points by just a tiny bit it'd probably throw off the entire image. I haven't used that function in a long time, but I remember trying to use the stacking feature back with CS5 and it did an absolutely terrible job.  I've heard it has gotten better, but everyone I talked to that did focus stacking all used third party software for that stuff.  Maybe there's a better program that people are using for this kind of work.

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