  We extend the work of Lai, Longhi, Suzuki, the first two authors and study
the 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 of
global fields $L/K$ that ramifies at a finite number of places at which $A$ has
ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for
the size $\delta_v$ of the local Galois cohomology of the Mordell-Weil group of
$A$ with respect to a $p$-extension ramified at a supersingular place $v$.
Next, in all characteristics, we describe the asymptotic growth of $\delta_v$
along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for
the change of $\mu$-invariants of $A$ from the tower $L/K$ to the tower
$LK'/K'$. Finally, we present numerical evidence supporting these results.
