The emigrant rosters on microfilm (from the FHL) cover up to the early 1900s. There are several films for the index. Check the FHL catalog under state of Utah for locality search. Scroll through the topics to find emigration. As the index is alphabetical, you will only need to order the film to cover the surname you are looking for. [I apologize to all for not citing the film numbers; the crossing the plains index is 0298440-42, and the crossing the ocean is either just before or after these numbers.] You can narrow the search by finding out what year they were first recorded in the census in Utah. Also, if you have birth dates/places for marriages of the children who emigrated and births of their children, you can narrow the window. The problem is that there are only a handful of Welsh surnames, so you really have to be careful to get the right family. Anyone who is researching Thomas, Jones, Lewis, Williams, Morris or Richards can tell you it requires some diligence. Large family history centers have the emigrant indices and ship roster microfilms in their permanent collection, so if you are near a large one, you should check before ordering. FHCs near temples are in this category. Here, in the Northern Virginia area, the Annandale FHC has the films. The HB Lee Library at BYU has the 2nd largest genealogical library, so that's an alternative to SLC. Especially if you have a student at BYU who can do the leg work. They have all the US censuses, too. Good luck.