Genes, whether dominant or recessive, do not dilute.
Actually, let’s start with the distinction between dominant and recessive genes. The distinction has nothing at all to do with how they’re passed on, only with how they’re expressed. At each gene locus, you have two genes. One of them came from your father, and one of them came from your mother. When you have kids, each kid will get one of your two genes from that locus, and one of the other parent’s two genes. Then, each of their kids will get one of those and one from your kid-in-law, and so on. At every stage, the person involved either has that gene, or they don’t. Nobody ever “sort of” has a gene, or half-has it, or the like.
Now, if the gene is dominant, then everyone who has the gene shows some trait associated with it. If it’s recessive, then you don’t show the trait unless you have two copies of the same gene. But even if it’s recessive and hiding behind a dominant gene, it’s still there, and will still be passed down (or not) just as easily as a dominant gene.
Now, then, suppose you have a mutation, that gives you a new gene, never before seen. Any given kid you have will be 50% likely to have it. If it’s a dominant gene, those kids will then show some trait, but even if it’s recessive, they’ll still have it; it’s just not doing anything yet. And maybe, just maybe, the gene will end up being more prevalent in the gene pool. This could be because it’s a dominant gene that gives some benefit, or it might just be due to luck (maybe the coin comes up heads for a larger-than-normal portion of your descendants, or maybe (either due to other good genes or due to more luck) you just happen to have more descendants than average). If this happens, then eventually, a couple of your descendants who both have your new gene will mate and have a child, and maybe that child will happen to get your mutant gene on both sides. If it was a recessive gene, this will be the first opportunity anyone has to see what it does, but either way, it gets passed down the same.