How do beets bake up in flour? I am thinking about incorporating some raw shredded beets in a vegan chocolate sheet cake. Since it is vegan, I will be substituting with Ener-G, chia, or flax egg substitutes and earth balance margarine, and can add more if a solution so dictates. In other baking, I have not found any need to make drastic accomodations to accommodate for moisture or texture. 

However, I am wondering if incorporating beets will alter the moisture levels. I will be balancing the earthy flavor with a lemon and lavender shortening-based frosting, so I am just basically looking for any confirmation that beets will not drastically add or leech moisture from the cake as I anticipate that they will function like carrots in carrot cake. 

If using them will change the baking process, on the other hand, I would like to know so that I can adjust the recipe accordingly. If so please include any tips to balance it out, bearing in mind that extra egg substitute or margarine is doable, but not if it entails one of the specific properties of actual eggs (feel free to include those too, bearing in mind that I am looking for a vegan solution).
I haven't tried baking with beets yet, but, as you said, it must be a bit like carrot cake.

To keep my carrot cake from going soggy while baking (carrots giving off steam), I mix them first with the recipe's sugar to weep them. This also pulls out flavour from the carrot to the cake batter. 

With beets, the procedure could be called bleeding rather than weeping, ha ha.