Although you wanted the physics question, I think it is important to answer the philosophical question. The question is: how can my own immaterial soul move my hand and body, when my hand and body are material?
The obvious answer is that the immaterial soul is attached to a material object, namely the brain, and it is some sort of high-level description of the enormous amount of computation going on in the brain. Computation is not a material object, but an abstract verb acting on data which is an abstract noun. It is equally abstract whether it is going on in a brain or in a computer. But in the case of a computer, you know exactly what is going on, so I'll just think about that. It is manifestly obvious to me that the computer is a good model of the brain, this was convincingly argued by Turing, but within philosophy, this is still a debate, which is why you can basically ignore philosophers. Their field is intellectually bankrupt, most likely because they have no arbiter of truth other than political persuasion: they have no theorems and they have no experiments. You can't make progress with pure politics. Historically, this method fails to produce true answers about questions as simple and obvious as "does the Earth rotate?"
The microsoft Windows operating system is just as abstract as a soul. It's an immaterial pattern, a collection of bits, which is realized in CDs, in computer RAM, in static RAM, or in magnetic drives. The windows soul is the same even if the substrate changes.
When you talk about cause and effect, you are always talking at a very high level. I don't know what the colloquial notion of "cause" and "effect" mean exactly in physical terms. They only have meaning in a computational sense. When you have a program running a factoring program, and you feed it 65537, you can ask what causes it to run through all possibilities without finding anything? The answer is that 65537 is prime. If you ask "why is 65537 prime?" the answer is because the program runs through all the possibilities! The notion of cause and effect is nonsense, it's not fundamental to physics, and it probably doesn't exist. It's a shorthand for "what can I change in order to have an impact in the world", and in this way of thinking, causality, as in X causes Y, is best formulated as "If I tell you I do X, is Y likely to happen?"
The notion that something abstract can interfere with the universe is either tautologically absurd or manifestly obvious. The properties of material objects will continue independent of any abstract realm considerations, so there is no way some agent outside of nature can change those things. But if there is a complex computation going on, the properties of these complex computations are revealed over time. So you can say "The primeness of 65537 made my program run through all the possible factors without finding anything". Somebody could then ask "how could the primeness of a number influence the physical world?" And the answer in this case is because this part of the physical world was computing something that depended on this property.
So if you view the abstract realm properly, it doesn't physically alter the stuff that happens to us, but it can provide a framework for qualities which are not apparent which explain the outcome of certain complex computations. If you formulate "abstract property A causes physical consequence X in system S" as "System S is implementing a computation where abstract property A implies consequence X", and then it is not mysterious how abstract properties can influence the physical world.
But some people say Jehovah came down and parted the red sea in a literal sense. This is mentally deranged.