This is the problem, with like... everything in existence.

It is not greedy behavior to refuse to handout money to someone who has no claim to it.

The easy way to solve this situation, even if you didn't want to get lawyers involved, and to do it "fairly", but still be providing for your children [which makes no sense to me- she is the children's mother, but does not have to ever provide for them financially in any way? even on the rare occasions they spend time with her?]---

She submits receipts to you for reimbursement.
If you proposed this to her you would get a lot of "oh that's not fair blah blah blah".

But it would be, completely fair.
And there would be proof that she is spending money on the kids, not on whatever the hell she pleases.

Frankly, if you get a lawyer involved and you have sole custody... she may end up paying child support.

I dont know nothin' about nothin'... but that makes way more sense.

you are not greedy because you resist giving away money.

No. Nope.
not at all.

wanting to keep your money does not equal greed.
Dont you believe it.

Also, I would puch for explicit and scheduled visitation and weekly scheduled phonecalls or something.
As someone previously mentioned- this randomness can be very damaging.