The most mentioned problem for dualism (see earlier posts for what dualism is and why it is important. Edit: By the way, what I mean by ‘dualism’ here is substance dualism) is the interaction problem. It goes like this: physical things and mental things have a different set of properties: physical things have volume, mass, charge, location, etc. Mental things, such as thoughts or minds, do not have these properties. If this is true, then how could a mind ever have an effect on the physical world? At best they would be epiphenominal, that is, causal floaters that can effect nothing. Thus if dualism is true there is no way for, say, my mind to move my body. But since our mental states do affect our bodies, dualism cannot be true. Our mental states either are physical, realized by the physical, or don’t exist in the sense we think they do.
My answer: The problem of physical causation is not specific to dualism. David Hume has famously attacked the notion of causation, showing that it is a meaningless concept if it refers to anything above and beyond mere regularities (given empiricism). “Necessary connexions” (his spelling) are leftover useless metaphysics, as they cannot be traced to anything we experience with the senses.
Later on the logical positivists were fine with this reduction of causation to regularity. But then if physical causation is mere regularity, why can’t non-physical things stand in regularity relations with physical objects? I see no reason why they couldn’t. That is, whatever answer anyone has to Hume regarding problems with causation work both for mental and physical causation. Other analyses (such as David Lewis’s counterfactual analysis) can also work with minds/thoughts too.
So there is no special problem for dualism regarding causation.