I know several young people are the ones in their family who have "done good" or "made it" -- they have a steady job and a stable place to live. Their siblings or parents may move multiple times a year, drive cars that keep needing calamitous repairs, and work off and on when they can. In these families, the one who is doing well looks after the others. That might mean putting someone in your spare room (or living room floor if you have no spare room) for a week when their spouse throws them out, or lending someone your car for a week, or "lending" someone rent that you know very well you won't see again. If you didn't grow up in a family that had this dynamic, I expect that to you this looks like prunable spending, equivalent to a daily latte habit or spending too much on cigarettes and alcohol. But from inside the family, be assured that this is not optional, and is part of the obligation of being part of the family.
Sometimes it's all one-way: I knew a young woman whose mother was a complete leech: she lived with them, paid no rent (used her welfare money for herself) and refused to watch her grand-daughter, so the young woman was paying her entire rent and daycare costs herself. But that's the exception. Usually in families like this the unemployed brothers show up to help you move or chop the firewood you got cheap because it wasn't split, people will care for your children or do an amazing job of organizing parties or whatever. There is a sort of family labour towards overall family goals. Some years maybe one person does a lot more or a lot less than others, because they are going through a hard time, or they had a run of good luck, or whatever.
So, knowing that, what you really need to discuss with your partner is budgeting in general. What you as a pair will spend on food, rent, gifts, vacations, furniture, transportation, clothes, treats and splurges, and, yes, money to family. You can work a number out together. If the family asks for an amount that is out of budget, you can say that. "Sorry, we just don't have it this month." But you can mostly say yes and provide an amount that, as a couple, you can afford. You and your partner can work out the tradeoffs -- more support to the family means a less luxurious vacation, or no eating in restaurants -- rather than just hoping the money will come from somewhere. If you want to save for a new car, or a downpayment on a house, you can see that this much a week to the family means this much longer until the savings target is achieved.
Your partner may decide the correct amount to budget for family support is zero. It's more likely they will not, but having a number (and sticking to it) will let you feel better that your other goals are still being met. Understanding how money and sharing works in your partner's family may take a while and require more listening. Perhaps your partner has told you why it's important to provide financial support, and what non-financial support the family has provided (or still does, or will) in the future. It's not as simple as "sending these people money is cutting into my latte budget so it needs to stop."