  Stability results for extremum seeking control in $\mathbb{R}^n$ have
predominantly been restricted to local or, at best, semi-global practical
stability. Extending semi-global stability results of extremum-seeking systems
to unbounded sets of initial conditions often demands a stringent global
Lipschitz condition on the cost function, which is rarely satisfied by
practical applications. In this paper, we address this challenge by leveraging
tools from higher-order averaging theory. In particular, we establish a novel
second-order averaging result with \emph{global} (practical) stability
implications. By leveraging this result, we characterize sufficient conditions
on cost functions under which uniform global practical asymptotic stability can
be established for a class of extremum-seeking systems acting on static maps.
Our sufficient conditions include the case when the gradient of the cost
function, rather than the cost function itself, satisfies a global Lipschitz
condition, which covers quadratic cost functions. Our results are also
applicable to vector fields that are not necessarily Lipschitz continuous at
the origin, opening the door to non-smooth Lie-bracket ES dynamics. We
illustrate all our results via different analytical and/or numerical examples.
