Can I get my mother's casket out of a cemetery and buried in another cemetery?

Yes

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A Grave Interest: Remains of the Day – Relocating a Body

agraveinterest.blogspot.com/2013/02/remains-of-day-relocating-body.html

Moving a body from one cemetery to another used to be a rare ... If the casket was placed in a concrete vault, it may be an option to move the ...

My great grandmother remarried, and she and her second husband bought a plot in the then new Holy Cross Cemetery in Colma, CA. They then exhumed her 20 year dead first husband (my great grandfather) and reburied him in their new plot, which had room for 20 burials. I find the whole thing grotesque, but you can do it if you are either the legal rep or get the approval of all her children

Certainly!

As long as you are listed as the contact at the first cemetery, sure.