The Halacha of "Sehita" (squeezing) forbids extracting absorbed liquid from a material on Shabbat. The question thus arises as to whether one may use baby wipes to clean an infant on Shabbat. The wipes have liquid absorbed in them which is then extracted when one presses the wipe against the child’s skin. Seemingly, this would be forbidden due to the Shabbat prohibition of "Sehita."
In truth, however, one who uses baby wipes on Shabbat has authorities on whom to rely, provided that he wipes gently and does not squeeze liquid from the wipe. If one looks carefully at a baby wipe he will notice that there is liquid even on the surface, which has not been absorbed into the material. It is certainly permissible to use that liquid for cleaning the baby, since it does not have to be extracted from the wipe. Therefore, one may clean a baby with moist wipes on Shabbat, provided that he uses only the liquid on the surface, and does not press down to extract the liquid that is absorbed.
For that matter, an adult who wishes to use moist wipes for cleaning himself on Shabbat may do so, provided that he does not press down to extract absorbed liquid.
By the same token, it would be permissible to moisten ordinary toilet paper with water and use it for cleaning oneself after using the restroom. Once again, one must ensure to wipe gently so that he does not extract any absorbed water. He must also ensure to use only a small amount of water, because if the tissue is saturated, water will inevitably be extracted during wiping. But as long as a person moistens the paper only slightly and wipes gently, he may use moistened tissue paper on Shabbat. Moistening tissue paper does not violate the Shabbat prohibition of Melaben (laundering), since tissue paper is disposable and nobody is careful to maintain its clean appearance. The only potential problem with moistening toilet paper is "Sehita," a problem which, as mentioned, one can avoid by using a small amount of water and ensuring to wipe gently.
Summary: One may use baby wipes or moistened tissue paper for cleaning on Shabbat, provided that he wipes gently such that moisture absorbed in the wipe is not squeezed out. One may moisten toilet paper with a small amount of water on Shabbat.