I had to sit down and figure out a very similar situation with my brother. It wasn't so much about taking his kids places, but he definitely took advantage of me in other ways (financial, emotional, etc). He was extremely manipulative and sly. I finally had to create my own boundaries for his requests. For example, I would no longer agree to help pay for anything like ski rentals, dance shoes, a better cell phone, but I would help out with what I consider essentials, like helping him keep his home phone turned on (because he had kids in school, and what if one of them got sick or hurt? I didn't want the nurse or school staff to be unable to reach him). I would not house-sit or pet-sit so he could go on vacation, but I would watch a child if another one had to go to the doctor or ER. Those kinds of things made sense to me. It's different if someone makes an occasional request, like they finally get a weekend away after a tough year, or they have a chronically ill child: in those instances I'd babysit or take their other children along with mine. But the chronically lazy, irresponsible, chaotic, draining people are another story altogether, and require a different approach.
So, I encourage you to draw your own boundaries. Perhaps you could care for one child when your sister gives birth to #4, to take some of the burden off your mom. Or when you're making cookies or cupcakes with your kids, you could make a batch for their cousins. Or your kids could make a summer craft for their younger cousins to do while their mom is resting. There are lots of fun crafts that your kids could prepare (cutting shapes, packing the necessary supplies like string or child-safe glue sticks and instructions in a zip lock bag). That shows family love and support without being the nanny and maid to a lazy person.
And now is a good time to explain your boundaries, and explain your family dynamic in simple terms to your children. Don't use words like "lazy" or appear to criticize your sister. Use words like "she's not always willing to do family activities like we are" or "they sometimes have trouble making plans and managing their family", and stress that this is private family business, not to be discussed, but if they need to ask questions or talk it over, they may, privately with you). You don't have to go into detail, but you can stress how you are responsible for your kids, and your sister and her husband are responsible for theirs, and are able to care for them. And while families can cooperate with each other and support each other of course, that doesn't extend to "oh, since you're going, take my kids too". You might use an example such as asking one of your kids "if every time you were invited to a friend's house, or you had a birthday, I insisted that you take your brother or sister along, or share your birthday, you wouldn't think I was making very good choices, would you?" And explain that some people need guidance and rules, but also stress that you are willing to help in a real time of need (illness, accident, job loss, childbirth, etc) but those circumstances are not the same as taking extra kids on every outing. Empathize with your daughter about overhearing rude words. Explain that those words hurt, even just to overhear them said to someone else, and hug your daughter. You can't change the words, but you can show your daughter that you care and help her learn to feel secure in her own family, with kind words and love.
Remember, you can't always change what others expect of you, and you may always be "expected" to subsidize your sister's laziness, but you can change and control how you respond to unreasonable expectations. Think through your boundaries, always be kind but firm, and step up in times of need and hold your ground calmly and politely in times of being asked to enable someone's weaknesses.
