Yes—this can be normal on the Z6. Nikon’s Eye AF on the Z6/Z7 was added later and is generally less robust than some competing systems, so it may jump between eyes and the face box or hunt before settling.

It tends to work best when the eye is larger in the frame, has good contrast, and there is enough light. It tends to struggle more with short lenses, wide apertures, lower light, flatter/overcast lighting, and low-contrast scenes—situations that make shallow depth of field and eye detection harder.

Things that can help:

• Fill more of the frame with the subject’s face/eyes.
• Use better light or higher-contrast lighting when possible.
• Stop down a bit for more depth of field, which can help on-sensor AF acquire focus.
• If it starts jumping, try re-engaging focus with another half-press.
• For difficult conditions, use subject tracking or a different AF method instead of Eye AF; some users find it more reliable.

So the behavior you describe does not necessarily indicate a fault—more likely a limitation of the Z6 Eye AF system under certain conditions.