You should get a court order to reduce stop child support obligations which will accrue against you if you fail to get a modification. Why you are leaving and quitting your job will be a big question. If the court does not think it is necessary, then it will impute income to you and order support in that amount. If you fail to pay, it will continue to accrue and the custodial spouse shall be entitled to a judgment. It does not matter if you lost/quit your job but failed to file for a downward modification.
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Order will not stop because you are moving. If the custodial parent stays in NY, then the order will continue. You should consider filing for a downward modification if your income is decreasing, even if it may not be granted if you are voluntarily moving.
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My colleagues are correct. Moving out of the country will not relieve you of having to pay child support, although it will admittedly be difficult for the custodial parent to collect. You should move for a modification of support before you go if your income will be substantially lower in the foreign country. If you do not, and you do not pay, then when you return, you could be facing a substantial deficiency amount and in some states, you could theoretically be charged with a criminal non-support action.
My colleagues are all correct. However, you did not indicate that you want to stop paying, or that you want the payments lowered, only that the payments can no longer be deducted directly from your employer. If you are going to continue to make the same payments, just continue to send them to the Support Collection Office in Albany.
If you want the court to modify the order to remove the payments from the SCU and make it directly payable to the custodial parent, you would have to file a modification petition. The other party would have to agree to such a change. Otherwise, the court will not do it.
Obviously, if you don't continue to make payments, you will incur arrears, have money judgments against you and, ultimately, have a warrant issued for your arrest. If you should ever attempt to re-enter the US, you will have big problems. The other issue is, are you going to continue to have a relationship with your children? You will have no ability to enforce any visitation rights. You will be at the mercy of the person to whom you are no longer sending child support.
Best of luck.