Does a neural network continue to change after SGD stops improving?
The evolution of the improvement of the performance of the network if you keep training it depends on:

1) the surface of the loss function with respect to the network parameters.

2) the learning rate, which is the scale factor you apply to the gradient in order to compute the next set of weights.

If the learning rate is too high (with respect to the shape of the loss surface) and you keep training, you may escape a local minimum you just reached.

The problem is that you don't know the loss surface and so we cannot choose the perfect learning rate for it. There are a lot of heuristic tricks to try to improve the exploration behaviour of SGD, like momentum.

So the answer is: it depends. The network could stabilize around a local minimum and so the weights would only suffer small oscillations, or it might ruin everything by escaping the local minimum.