If you travel with kids that are not your own, you need a written confirmation of the parents / the legal representative of the child. It is not sufficient to have an oral agreement! Please make sure that you follow the regulations for your home country and all the countries you are crossing. You can consult the German embassy for details. Here are some regulations for German parents that mind be adequate for you as well.
You should ask the legal representative of the child for an “Übertragung der Personenfürsorge” (a transfer of personal welfare) for the child in question. These are similar to a form you need to accompany an underaged person to a place where only adults are allowed, for example a disco. For travelling with children you need more information, the document you and the parents are signing should contain these topics:
First, information about the accompanying child aa first name and family name, place and date of birth, home address, the passport ID and the place and date that the passport was acquired. Second you need some data of the parents, as first name and family name, the passport ID and the signature. All people legally responsible for the child have to sign your confirmation /normally both parents)! You should add copies of the passports of both parents. In addition, you need a telephone number of the parents with the appropriate country code. Finally, you have to write down the data of all people responsible for the children during the journey including their addres, and the terms of the journey, the route you are taking and the final destination. The confirmation should be written in the language of the country you are travelling to, at least you should have an English version.
Remember, these are regulations for German parents as an example that might not fully cover your needs!
You should be prepared that the child might get ill during the journey. Please take an appropriate amount of the medicine you might need with you. You must ask the parents about special medical issues that might apply. You should be especially aware of allergies and vaccinations the child has. You should have a special allowance to let the child be medically treated if necessary. Please note: Some countries demand a given kind of confirmation or even a notarial fixed commitment. Ask the embassy for advice.