Why does time-domain convolution correspond to frequency-domain multiplication? (visual)

I seek a visual explanation of this. I’ve already seen the maths, and can derive the proofs – they amount to nill for an intuitive understanding. Any amount of math is welcome, as long as serving to ultimately explain it visually.

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Examples of excellent explanations:

• Fourier Transform: the "winding around the circle".
• Convolution: the "input-side algorithm", and how it ties to output-side.

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Continuous vs Discrete: both are welcome, but ultimately the discrete case must be explainable. I’ve long thought of the input-side algorithm myself for continuous convolution, but never completed the picture; real analysis is tricky.

Circular vs linear: a complete explanation ought to cover both, but I’m primarily interested in linear convolution (rather, how the circular of padded signals is equivalent to linear).

Duality: ideally, should be covered (conv in freq domain <=> mult in time domain).

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Ideas:

1. Only right-padding works, and both inputs are padded; this looks like forcing inputs to correlate with lower and fractional frequencies (and more frequencies) relative to unpadded’s frame.
2. Something about convolving with shifted deltas and observing modulations of complex sinusoids in other domain; awaiting clarification from @AndyWalls.