I spent my entire childhood being falsely accused of things by my NM, from stealing her jewelry, to raiding the liquor cabinet when my parents were out, to . . . you name it.
To this day, I'm not entirely sure if M believed the accusations, or if she made them up as a way of sabotaging me and turning my father and siblings against me. The accusations became so constant, that by the time I was in high school, my father was believing them, and also making false accusations. I remember one afternoon, sitting in my room doing my homework, and my father burst in and accused me of smoking pot in my room. He had NO reason to suspect that I was smoking anything, but M had been constantly accusing me of smoking both pot and cigarettes (I had never smoked anything), and he finally started to believe it. I'm just not sure if M believed these accusations, or was trying to gaslight me, or both.
Bottom line, the only way I could end it was to go NC. I still get letters accusing me of things, but at this point I ignore it. I would have to say, that of all the abusive things she did to me, the false accusations and sabotage were probably the worst.