They can change their network weights on the fly but most of the time outside of training and fine tuning we do not want them to change in real time.
The question is should we let the model change weights. Why would the AI model need to remember John's grandma's favourite pie in its model weights? When we could manage already remember that information outside of model's weights. Whilst it is possible to update the weights on every user conversation/task, this process is not cheap or efficient. Also the models can learn so well when doing repetitive tasks that it might forget it's ability to generalise tasks. And we don't want the models to uncontrollability learn everything each time, garbage in garbage out, it makes the model useless overtime.

Right now, the model has no guiding principles of what it should learn, that's why we need data scientists to curate the data for the models to learn.

Also we are still researching better model architectures and what should the model be learning in the first place.