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We extend the work of Lai, Longhi, Suzuki, the first two authors and studythe change of $\mu$-invariants, with respect to a finite Galois p-extension$K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension ofglobal fields $L/K$ (whose characteristic is not necessarily positive) thatramifies at a finite number of places at which $A$ has ordinary reductions. Weobtain a lower bound for the $\mu$-invariant of $A$ along $LK'/K'$ and deducethat the $\mu$-invariant of an abelian variety over a global field can bechosen as big as needed. Finally, in the case of elliptic curve over a globalfunction field that has semi-stable reduction everywhere we are able to improvethe lower bound in terms of invariants that arise from the supersingular placesof $A$ and certain places that split completely over $L/K$.

The $\mu$-invariant change for abelian varieties over finite $p$-extensions of global fields