I have a simple 3d model with two animations created in Blender. After successful exporting I noticed that position of bones was changed and I have no idea why. When I try to import mesh with skeleton from exported babylon file it has this weird position and it seems like animation lost half of frames. Also some of vertices during animation stick to one position
For the frame loss, the frame rate of Blender needs to be either 30 fps or 60 fps (30 will result in much smaller file). I have my own animation system / exporter, so I never needed to figure out where that went in the Blender 2.80 version. @gryff, do you see it?
Some vertices not moving points to a armature weight issue. Did you parent the mesh to the armature with automatic weights? In any case, you should probably visually check where the weight of each bone is being placed. Probably some vertices have no weight from any bone.
Select mesh, & change to weight paint mode.
With the mesh data properties visible, select each vertex group, and note the vertices being painted.
Here is a snap I did a while back. Red indicates high weight, dark blue none. For a couple bad areas, just paint with the bone which should be influenced by that bone. This exercise should probably be done as a matter of course.
Maybe it has something to do with your Walking animation having nonzero location and nonzero xyz rotation in Pose mode, whereas your Idle animation seems okay (with zero location and zero xyz rotation).