  We deduce stability and pathwise uniqueness for a McKean-Vlasov equation with
random coefficients and a multidimensional Brownian motion as driver. Our
analysis focuses on a non-Lipschitz drift coefficient and includes moment
estimates for random It\^o processes that are of independent interest. For
deterministic coefficients we provide unique strong solutions, even if the
drift fails to be of affine growth. The theory that we develop rests on It\^o's
formula and leads to $p$-th moment and pathwise $\alpha$-exponential stability
for $p\geq 2$ and $\alpha > 0$ with explicit Lyapunov exponents, regardless of
whether a Lyapunov function exists.
