I am keenly, keenly aware that all this is easier said than done, and that I may eat my hat when this moves from theoretical into the realm of reality.
But my plan for any relatives that want assistance in the future is that first and foremost, if you want significant money from me (more than a $100 gift card at Christmas type money, or maybe a one time lump sum to deal with an unexpected expense), yes, your finances do become by business. If you are claiming dire circumstances, then the first thing to do is to get the budget in order. Just like we want a full case study in MMM, I need to do a full case study of Relative's finances to help him or her determine what changes can be made.
Second, they must be willing to make those changes. I'm not going to give $400 a month when there is a $120 cable bill. Once you start spending my money, frivolities go. I'm not suggesting it has to be a completely bare bones life; that's not what I'd want for someone I care about. But MMM level cuts? Yup!
Third, before I give any money, any entitlements will be maximized. SNAP, elderly care programs, Meal on wheels, disability--whatever they qualify for. It's not reasonable, IMO (even if it is understandable) to want to spend my money because one is too proud (or too lazy) to take advantage of any programs available to them.
Fourth, most of my assistance will not be in the form of money. It will be helping to find a realtor to sell the house and then locating a decent, cheap rental in an okay area. It will be helping to rehome pets if they are an unreasonable barrier to rental. It will be working with them to apply for jobs. It will be assisting with locating and applying for the aforementioned entitlements and programs. If all of this is too much work, then clearly the need is not as dire as s/he is making it out to be. It can't be more important to me that it is to them, and they can't just decide my money is a better solution than all those other things. And if there are mental health issues, I will support in any way I can--including paying for care--addressing those issues.
Fifth, there will be ongoing reviews of finances, and if irresponsible stuff is happening, my checks will stop coming. If suddenly credit card bills start appearing and cable is reconnected and there's a lovely new sofa in the living room, I'm done.
I'm not going to jeopardize my FIRE plans for someone who, even when things are supposedly desperate, can't be bothered to make many of the sacrifices I've made for years. And I may sound like a complete asshole about all this. That's fine. In a case of true need, I would have no problem providing money, even lots of it. But when it's self-created need and the person isn't willing to take the help or make the changes necessary to get things turned around, I would consider it throwing good money after bad. When you give $400, an irresponsible person is going to instantly adjust the lifestyle to that amount, and then in no time at all it will be $500, or it will be an emergency expense that required a large extra lump sum because every penny of that $400 was spent every month. And thus, their life won't actually be happier or more stable, so the only real change will be that I'm poorer.
It's a crappy situation, and I'm so sorry you are having to ask these very difficult questions, especially when there are no good answers.