When you transcode from 8 bit to 8 bit you will almost always have some issues with banding if there are any changes in the gamma or gain within the image. As you are starting with 8 bits or 240 shades of grey (bits 16 to 255 assuming recording to 109%) and encoding to 240 shades the smallest step you can ever have is 1/240th. If whatever you are encoding or rendering determines that lets say level 128 should now be level 128.5, this can’t be done, we can only record whole bits, so it’s rounded up or down to the closest whole bit. This rounding leads to a reduction in the number of shades recorded overall and can lead to banding.
DISCLAIMER: The numbers are for example only and may not be entirely correct or accurate, I’m just trying to demonstrate the principle.
Consider these original levels, a nice smooth graduation:

128,    129,   130,   131,   132,   133.

Imagine you are doing some grading and you plugin has calculated that these are the new desired values:

128.5, 129, 129.4, 131.5, 132, 133.5
But we cant record half bits, only whole ones so for 8 bit these get rounded to the nearest bit:

129,   129,   129,   132,   132,   134

You can see how easily banding will occur, our smooth gradation now has some marked steps.
If you are rendering to 10 bit you would get more in between steps.
If you render to 10 bit then when step 128 is determined to be be 128.5 by the plugin this can now actually be encoded as the closest 10 bit equivalent because for every 1 step in 8 bit there are 3.9 steps in 10 bit, so (approximately,translating to 10 bit) level 128 would be 499 and 128.5 would be 501
128.5 = 501

129 = 503

129.4 = 505

131.5 = 513

132 = 515

133.5 = 521

So you can see that we now retain in-between steps which are not present when we render to 8 bit so our gradation remains much smoother.