As a techie - not a guitarist, the suspicion would be that the strings are seated a bit high in the nut, or that the frets are the "large size" that stick up a bit more than you're accustomed to. You may just need to adjust the neck, to get the strings a bit closer to the fingerboard.

When the capo pulls the strings down against the fret, it has to slightly stretch the string. If you're setting the capo on the fingerboard, just behind the fret, there's a bit of additional stretch.

Usual advice from the gitterists is to be careful to get the capo on the fret or as close as possible, to minimize the extra stretch in the strings.

The amount of pitch change for a given amount of additional length will vary slightly with the tension of a given string relative to it's intended "working load." Usually, a "slack" string will change more than one that's close to normal tension. Changing to a heavier string on the ones that are the most problem, so that they're tuned a bit tighter, may reduce the "de-tuning;" but won't make it go away completely. It will increase the load on the neck and may not suit your playing style.

John