The speed of the soundwave coming towards you will not change even if the lower frequencies are more omnidirectional so there will be no shift.
The SPL of these different frequencies will change though depending on distance because of this effect.
That’s why low and mid drivers ‘need’ reinforcement from the walls in a traditional loudspeaker arrangement.
So speakers are made to be listened to at certain listening distance. Wall reflections also have a large influence on the sound because they receive mainly off-axis sound which has less high frequency content.Rgs, Collin