Tax question....sisters ex hubby claims their children?

I had posted a question about my sister who has been legally seperated from her husband for about 3 yrs. He claims two of their children and his brother claims the other two. I was asking about the legality of it. My sister and her children have lived with my mother since may or june. My mother was going to claim them because my sister doesn't work. I was told that if my mother sends in a paper return the irs will ask for proof as to which had them. What constitutes as proof? In Aug. my mother was evicted because of my sister and her kids living with her the landlord knew when she moved in. The reason she was evicted was because my sis was to move out and never did. Since that time I have given them an apartment in the building my hubby and I own and they live together. Is that proof? or what? (Also, the schools the kids have attended and daycares no theyve lived with my mom)