The basic position properties for the Grid emitter and other types of emitter, sets the properties of the birth position of the particles, animating these properties will affect the particles that are born in a later point in time, to move the particles you can use one of the many tools that are available , e.g. Transform node  , Motion node.

even with the transform node it is the same, the grid stays at the first keyframe. I have linked the emitter position XYZ to the position of a light. but this should not be the reason because the particles dont move with the transform node, right?

For a specific setup please send us an email to support with the affected comp.