Mosses are able to transport water and nutrients internally through intercellular voids and cell-to-cell connections involving exchanges of cytoplasm, as well as externally across their surface. Labeling mosses as nonvascular plants is misleading because mosses in the Polytrichales order have highly specialized water and nutrient conducting systems which are reminiscent of xylem and phloem. But in general, a moss's main vascular system can be said to be its outer surface which is carefully designed to store and conduct water across the plant.
External Conduction
Moss cells absorb water and nutrients directly from their environment, so many of their nutritional needs can be met simply by ensuring water and its dissolved nutrients can spread across the entire surface of the moss plant. The water is able to spread through capillary action, aided by a carefully designed surface. Moss leaves are frequently appressed to the stem and have sheathed bases which create small places for water to climb along a stem. The leaves may also be covered in small bumps (papillae) which create capillary spaces. Rhizoids, which are somewhat like roots except typically only 1 cell thick, can also make good paths for water to flow across. Some mosses have their stems covered in rhizoids or paraphyllia to help conduct water. Sphagnum mosses have large cells across their leaves which exist mainly to absorb water like a sponge, and Leucobryum mosses have a layer of water holding cells on both sides of their leaves. Colonies of mosses often form compact mounds that allow for water to flow across the entire mound.
As far as I know, moss cells do not deliberately dump nutrients into the film of water on their surface in the hopes that it will be absorbed by other cells of the plant that need it more. But, when dessicated moss is rehydrated, solutes in the protoplasm may leak into the external film of water. These contents might be able to be reabsorbed by other cells of the moss, forming an incidental external nutrient transport pathway.
Internal Conduction
Water and nutrients are also able to flow through cells or spaces between cells from inside the moss. Experiments have shown that nutrients are relocated from old growth to new growth using cytoplasm-linked pathways in Sphagnum and Polytrichaceae mosses. Many moss stems and almost all seta (the stalk connecting a spore capsule to the main body of the parent moss plant) have special nutrient conducting parenchyma cells surrounding water conducting cells called hydroids. The central ribs of moss leaves also often show a similar pattern. Sometimes these chains of nutrient conducting cells in the leaves connect to the main conducting strands in the stem, but usually the leaf conducting strands come to an end somewhere in the stem cortex without connecting.
Mosses in the Polytrichales order have the most specialized nutrient and water conducting cells of all mosses. The nutrient conducting cells for mosses in this order are different enough from other mosses to be given a special name--leptoids. 874
The vascular system in these mosses are not evolutionarily homologous to xylem and phloem, but serve the same basic transport purpose. In Polytrichum moss, water has been measured to flow 200 cm/h through the stem and organic compounds at 32 cm/h. These specialized hydroids and leptoids are what enables the tallest moss, Dawsonia superba, to grow up to 70 cm.
The water and nutrient transport architecture for mosses outside the Polytrichales is more variable and less specialized. Andreaeopsida, Andreaeobryopsida, and Sphagnopsida completely lack hydroids. Takkakia's hydroids are unique in that they are short instead of tall and transfer water through pores formed from plasmodesmata at their ends. Sphagnum has nutrient conducting cells, but they are not homologous to the nutrient transferring parenchyma cells of other mosses. Some mosses in families which typically have central conducting strands have lost that feature over time, and mosses which would normally have hydroids may fail to develop them if grown underwater.
Outside of the usual stem, seta, and leaf midrib conducting cells, it has been suggested that rhizoids and caulonemal cells also conduct nutrients due to similarities in cell organization to the nutrient conducting parenchyma cells. The cells that form the interface between the parent gametophyte and the sporophyte foot also transfer nutrients.
Sources
<PERSON>, R. N. Biology of Bryophytes. 2005. Chapter 11 "Conduction in Bryophytes".
<PERSON>, B., and <PERSON>. Bryophyte Biology. 2009. Pages 60-70, 83, 301.
Ligrone, <PERSON>, and <PERSON>. Conducting tissues and phyletic relationships of bryophytes. 2000. 874