One line of evidence for ion transport comes from macroscopic examination of statistical ensembles.  All these techniques use intact vesicles with an entrapped volume, with ion channel activities reported by different spectroscopic methods.
In a typical case, a dye is entrapped within the population of vesicles.  This dye is selected to be respond colorimetrically or fluorometrically to the presence of an ion; this ion is typically absent from the inside of the vesicle but present in the outside.  Without an ion transporter, the lipid bilayer as a kinetic barrier to block ion flux, and the dye remains "dark" indefinitely.
As an ion transporter allows ions on the outside to diffuse in, its addition will affect the color/fluorescence property of the dye.  By macroscopically monitoring the dye's properties over time, and controlling outside factors, the ability of a compound to act as an ion transporter can be measured.
Observing ion transport, however, does not pin down ion channel as the mechanism.  Any class of transporter can lead to the same observation, and additional corroborating evidence is usually required.  Sophisticated experiments intended to probe selectivity, gating, and other channel parameters have been developed over the past two decades and recently summarized.