Yes, sometimes. But, abdominal movement isn't necessarily a foal.
I've seen too many mares that folks purchased as being 'in foal', with the seller saying, "Look! You can see it moving in there!", only to palpate the mare after they got home with her and find that she's not bred.

I very rarely see foals moving or kicking inside the mares.

When I do, it seems it's usually right after the mare has eaten, or when she's standing around lazy. If you watch long enough, sometimes you'll see it, generally at the back of her belly, towards the stifles.

Usually the mare will act uncomfortable about it, too.

But I work with pregnant mares, and I rarely see it (I also never have time to stand around and watch them).

Maybe that's why so many people have bought a mare with a "haybelly" only to find a surprise in the stall one fine morning.

Oppie was obviously pregnant and VERY uncomfortable towards the last couple of months. She liked her belly rubbed too.