Let \mathbbP_n *K_hn(x) = n^-1h_n^-då_i=1ⁿK((x-X_i)/h_n){\mathbb{P}}_{n} \ast K_{h_{n}}(x) = n^{-1}h_{n}^{-d}\sum_{i=1}^{n}K\left((x-X_{i})/h_{n}\right) be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law \mathbbP{\mathbb{P}} on \mathbbR^d{\mathbb{R}}^{d} . It is shown that the processes f ® Önòfd(\mathbbP_n *K_hn-\mathbbP)f \longmapsto \sqrt{n}\int fd({\mathbb{P}}_{n} \ast K_{h_{n}}-{\mathbb{P}}) , f Î Ff \in {\mathcal{F}} , converge in law in the Banach space l^¥(F)\ell ^{\infty }({\mathcal{F}}) , for many interesting classes F{\mathcal{F}} of functions or sets, some \mathbbP{\mathbb{P}} -Donsker, some just \mathbbP{\mathbb{P}} -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.