Why multiplying frequency kernel with sinusoid results shifting frequency domain
This can be explained by the Convolution theorem

As you state, in the time domain you have the kernel (f) multiplied by a sinusoid (g).

If we take the Fourier transform of f·g (i.e., F(f·g)), this is equivalent to the convolution between F(f) and F(g) (i.e., F(f)*F(g)). Note that the Fourier transform of the sinusoid is proportional to two delta functions located at the frequency and negative frequency of the sine wave. So when you convolve these deltas with the frequency kernel your frequency kernel "shifts" to be centered around the sinusoid  frequency.