A Delay Between Two Filtered Chaotic Signals
I assume that your experiment where you saw no effect was with signals that underwent a linear phase vs frequency process in their delay (constant group delay). Under many situations the phase of the channel in the frequency domain is non-linear, causing different frequency components to experience different delays (group delay variation). When you frequency select portions of such signals, you will then see the result of the delay that is dominant for those bands.

This is the same reason it is not best practice to simply use cross-correlation peak to evaluate the group delay. See further details specific to the challenges of doing this, and the better solution for those cases, at this posting here:

How determine the delay in my signal practically

And here:

How to reliably compute the group delay of a comb filter

At the bottom of the first post is a plot of group delay that illustrates my first point. This is an extreme case of delay variation where we see as duplicated below there are frequency selective regions where the delay is as much as 15 samples, while generally elsewhere it is only around 2-3 samples. If we further filtered this signal (with a linear phase filter to not cause further delay variation), we change the weighting between a 15 sample delay and a 2-3 sample delay.