Hi, there! I'm not completely certain I follow the scenario. Mitochondrial DNA is passed along exclusively from the mother. We all have mitochondria, but they come to us only in the ovum.
So you and your discovered 1st cousin (since you're 10 years apart in age, the only other feasible relationships for that amount of shared DNA would be great aunts/uncles or great nieces/nephews) would share the same mtDNA only if your mother is the sister of her mother. If the connection is through your uncle, the mtDNA would be different.
Works the same way for prior generations, too. Just one male anywhere in the inheritance chain breaks the mtDNA continuity. And because most western surname traditions are patrilineal, it often makes it very difficult to accurately determine the mtDNA inheritance sequence on paper.
Too, because autosomal DNA is divvied up 50/50 at conception, any pedigree collapse along the way could account for atDNA sharing and different mtDNA, even if the matrilineal paper-trail is rock solid. For example, if your 2g-grandparents were cousins, you might expect to see some of your 2g-grandfather's DNA showing up in some of your 3rd cousins who look on paper like they should be only on your matrilineal line.
And last, you didn't indicate how distant the other shared matches are. It's in the testing companies' best interests to make it seem like a piece of cake to reach back to your Xg-grandparents via our inexpensive microarray DNA tests. In truth, things get really iffy past 4th cousins, or 3g-grandparents. It requires significant understanding of DNA, multiple test takers, and a good deal of legwork--things like comparing the shared segment to population and haplotypic pile-up regions; evaluating viable SNP density in the segment; traditional trio phasing if possible; determining if the segment resides in the exonic region, which is less reliable for genealogical comparisons; etc.--in order to arrive at an accurate assessment. If any of those distant matches are showing very small segments, they may just be muddying the water; they may not be actual evidence of a genealogical relationship.