When they first started placing the Niftar in wagons to take them to the Bais HaKvaros, the gedolim raised a ruckus, since Kavod HaMeis dictates that a Niftar should be carried, just like the Aron Kodesh was carried and not driven in a wagon. Rav Moshe Shternbuch (2:617) says that automobiles are only used when needed to traverse longer distances to the cemetery.
At the very least, says Rav Shternbuch, the driver of the automobile should be a Shomer Torah U'Mitzvos and not a goy or michalel shabbos. He brings proof from Yaakov Avinu who gave clear directions as to who could carry his Mittah.
Although when transporting a Sefer Torah we are not particular about having a Jewish driver, that is because the Sefer Torah must be held by someone, making the driver irrelevant as the one carrying it is giving it the honor it deserves. Whereas a mittah is not carried in the automobile by those accompanying it, but rather lying in it, rendering the driver the one doing the actual transport.
Although having a goy drive is not an issur, Rav Shtenbuvh says it is worthwhile to try to arrange for a Shomer Torah to drive in order to give honor to the Niftar.
Important Note: We bring this tshuvah as a starting point for discussion and not to convey any halacha. We try to convey the Tshuva to the best of our ability. We admit that our understanding may not be accurate. One should learn the tshuva to verify the accuracy of our interpretation. Please understand that this Tshuva may not be the final word on this topic. One should consult a Rav before drawing any conclusions.