There are many ways that the income tax dependency for children in divorce can be handled, although most of them are not supported by the law. If the parties are in agreement, then nearly any method of claiming the children can be used, including alternating years, splitting the deductions between the parents (in the case of multiple children), allowing the non-custodial parent to claim the children provided that child support is current, a combination of these, or any other method that the parties can come up with.
However, under Missouri law, the custodial parent is entitled to claim the children for tax purposes each and every year, period. So unless there is an agreement otherwise, this is what the Court will order. However, regardless of any agreement or Missouri court order, federal income tax law requires a child to reside with a parent at least 50% of the time to be claimed as a dependent. So, if the parties have agreed to something else, such as alternating years, and the non-custodial parent has the children less than 50% of the time, the Internal Revenue Service (or Department of Revenue) can reject the claimed dependency notwithstanding the court order, as Missouri family courts do not have jurisdiction over the IRS or the power to alter federal law. The only remedy would then would be for the parties to return to state court to seek reimbursement for the dollar value of the claimed dependency from the other parent, which the Court may not even entertain.
So the general rule would be that the custodial parent gets to claim the children each year, unless an agreement is reached otherwise. But the word of caution on agreements (particularly for the non-custodial parent), is that both parties have to follow it, or it can be rejected by the taxing authority with the only possible remedy being a return to state court.