The major breakthrough in this was made by Maryam Mirzakhani in her thesis (see this preprint) Her stuff is is relevant to general intersection theory on moduli space (see references to Witten's and Weitsman's papers in her bibliography), which is one way to interpret your question about "not top-dimensional classes"). Her work has received a lot of attention (Google Scholar shows 83 citations) -- I am sure those 83 papers have a lot more information.

The major breakthrough in this was made by Maryam Mirzakhani in her thesis (see this preprint) Her stuff is is relevant to general intersection theory on moduli space (see references to Witten's and Weitsman's papers in her bibliography), which is one way to interpret your question about "not top-dimensional classes"). Her work has received a lot of attention (Google Scholar shows 83 citations) -- I am sure those 83 papers have a lot more information.