ag.algebraic geometry – Ascend and descend properties for arithmetically Cohen-Macaulay/Gorenstein varieties

I had few questions regarding arithmetically Cohen-Macaulay and Gorenstein varieties. A projective variety is called arithmatically Cohen-Macaulay/Gorenstein if its homogenous coordinate ring is so. I was wondering whether any of these properties ascend along faithfully flat finite morphisms, i.e. if $$Yrightarrow X$$ is a faithfully flat finite morphism where $$X$$ is such a variety, is $$Y$$ the same type of variety? How about descending property? Does any of these properties descend along faithfully flat finite morphisms?

Lastly I was wondering whether there are any nice examples of families of varieties that are arithmetically Gorenstein? Preferably with dimension greater than $$1$$.