Let's suppose you are Ardor forger and you have an option to include a block (for free since it is your turn to forge) and get some childchain tokens or do not include and miss those tokens. So it is an option to get some value VS an option to not get any value. Which one do you choose?
It depends on the cost of processing those transactions. I don't know what that will be. You'll have to receive and store them eventually regardless, when someone else includes them, but presumably there will be some overhead in doing it in advance. You may not want to devote resources to this, especially if you are in a race with other forgers, or if you are on a device with limited processing power, memory, or bandwidth. Also, block-chain space may become a limited resource, in which case you'd rather include high-paying transactions than ones from low value child-chains.
Also, I think child-chains will be sponsored one way or another quite often. Although some may function as private currencies, more will treat their tokens merely as fuel for the network, and all the attention will be on things like property rights and voting rights that are represented in the network. A business will probably want low fees, so their users aren't struggling to fund transactions, even if it means paying all the Ardor themselves. Also, if space in blocks becomes contended, then then may want to pay higher Ardor fees to get their transactions verified quickly, without their users having to think about it. In general, the value of the child-chain to the owner is not the market capitalisation of its tokens.