It sounds like you have goldenrod under the tree.

It blooms at the same time as ragweed & gets the blame for making people sneeze, but really it's the ragweed's fault.

I don't disturb goldenrod in my garden.
It makes a beautiful, brilliant yellow punctuation mark when everything else is ho-hum boring.

I wouldn't bet on anything voluntarily spreading toward shade & away from sun;
most plants would migrate toward the sun, out from under the tree canopy.

Beneath & around my weeping willow right now, there's some tough coastal bermuda grass (the stuff they bale here in Texas to make hay!) & a few Mexican Primroses.

Little by little, I'm weeding out the bermuda & leaving the wildflowers.

Once every summer, I cut them back, & they always bloom the following spring.

Any area not colonized by the Mexican primrose can remain bare, covered with a thin layer of mulch.

If you don't like the look of mulch, & you don't want Mexican primrose, you might use English ivy, vinca, or some other low-maintenance, shallow-rooted groundcover.

Coleus is beautiful & colorful, but this is the first year I've used it:
I'm *thinking* it freezes in the winter & you have to take cuttings & keep them on a sunny windowsill all winter & then re-plant it in the spring.

Best luck!

sylvia

(ps:I think we gardeners fret ourselves silly over what'll grow under a tree:
the tree doesn't want anything under there drinking up its water, & most plants don't want to be under the tree, anyway.)