The worst problem (very common) is that bands start an up-tempo tune quickly and then, as the performance develops, gradually slow down. The result is that the music begins to drag and sound weary. I think the reason for this may partly be that so many musicians are growing old and have lost the vigour they once had. But I wish they would be aware of this and take more care.
The reverse sometimes happens: a tune speeds up as it is played. This can put one or two of the players into difficulties.
However, speeding up is by no means as bad as slowing down and can even be deliberate and exciting, especially if building to a special 'out' chorus. The Ken Colyer Band used to be noted for this and they themselves described it as 'controlled acceleration'.
With slower tunes, such as many ballads, there is less of a problem, though I sometimes find bands take a tune too slowly and it begins to drag.
When there is a singer, it is important that the tempo should be one the singer is comfortable with, so it helps to ask the singer to give an indication of the tempo desired or even to count the song in.
As in so many aspects of traditional jazz performance, that great young band Tuba Skinny are setting an example to us all. Notice how much trouble they take to get the tempo right. This is often done with much foot-tapping before the tune begins, while they (especially Shaye the cornet player) test the tune inside their heads just before starting; and they always keep the tempo under control throughout the performance, with rigid discipline from the rhythm section.
This aspect of their playing rewards study. For a typical example (and a good tune - Deep Henderson),
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The Book Playing Traditional Jazz is available from Amazon.