When more than one parent is eligible to receive child benefits through their employer, the matter of which benefits you receive depends on a few factors.
If your family (the kids in particular) resides in one of the cantons in question, you are obliged to receive your child benefits in that canton. If neither parent works in the canton which the children reside in, child benefits are paid through the employer of the parent which earns the highest salary. If one parent is employed and the other parent is self-employed, the employed parent receives the child benefits. If both parents are self-employed, the parent with the higher income receives the child benefits.
When the child benefits offered by the canton in which the non-eligible parent works are higher than the benefits in which the parent who receives the benefits works, you can apply to have the difference made up.
Example: You work in the Canton of Zurich and earn a higher salary than your spouse who works in Zug. Because of this, you receive your child benefits through your employer. However, because your spouse works in the Canton of Zug, they can claim the difference between the child benefits in Zurich and those in Zug.