Hello,
I don't know what type of filters you are designing and what you mean with "response time", but some things can be said.
When you think of the time it takes before the output of the filter is stable, this delay time is proportional with 1/BW, and some other factors (like allowable envelope distortion).
So a filter with a Center frequency = 100 MHz and BW = 1 MHz, will have the same delay time as a filter with Center frequency = 1 GHz and BW = 1 MHz.
You may know that the faster the envelope/frequency/phase of a sinusoidal signal changes, the more spectrum that signal occupies. So a narrow band filter will only pass the low frequency sidebands, hence the output envelope (and frequency or phase of output) will vary slowly.
When you understand the behavior of low pass filters, a band pass filter can be seen as a shifted (in frequency) version of a low pass filter.