I have bought Battle Chess from Gog, but there is a problem i am having. The animations and movements in the first Battle Chess is very very slow and frustrating. I wonder if it is a problem about windows compatibility or is it just the way the game originally was?

I remember the animations not pausing during the sound when I played it as a kid as well. I wonder if it had to do with the fact that we all had dedicated sound cards back then, and the game was programmed to do sound in parallel...

IIRC, the EGA version didn't have any movement sounds, only battle sounds -> faster movement animation.
The VGA version has movement sound effects -> slower animation.

This version is very slow and the sounds are not quite correct. I used to play this on an Apple IIGS and still can with an Apple IIGS emulator and it is much faster but the graphics seem to be the same

The sounds on an Apple IIGS would not be the same as on a DOS PC simply because the hardware was different. That does mean they are incorrect, just different. The graphics would have been very similar however.

It is not solved - the sounds is slowing the animations. GOG should better use some other version of the game.

battle chess was released in 2 versions. the version released on gog is the second one, with improved graphics and added startup music. the version some of you might remember is the first one. that version might be faster. the amiga version of course was built to run on the amiga hardware, which is better than the old dos supported hardware. i have played that too.

What a condescending terrible answer to your customer.
That's not it. The animations are very choppy because of the sounds. Not sure what the issue is but the original game on original hardware was not choppy like this.
If you disable the sound, the issue is gone and the animations are much smoother. Slow, yes, but not choppy.
I'm guessing some improvement could be made to the dosbox settings but haven't fiddled with them.

Try a Dosbox fork like Dosbox Staging or X. Actual DOSbox hasn't updated seriously since Barak Obama was in office. And GOG uses that version. Unlike Staging, whose latest release candidate was January 21st, this year.

Chances are, they've fixed whatever the issues are, given that they implemented proper "as a DAC" support.