All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
In principle, it is impermissible for a wife to leave her husband's house without his permission. In case of a revocable divorce, it is impermissible for her to leave her house except for a necessity.
Therefore, if your mother-in-law is still married to your father-in-law or if the divorced is revocable, then it is impermissible for her to leave the house, let alone travel, unless there is a necessity.
However, if the divorce is irrevocable or her husband gave his permission, then there is no problem in her traveling with her son or living with him. The fact that she spoke ill of him, or that he mistreated her, or that she caused friction between him and his siblings, does not prevent her from living with her son or traveling with him. But she must repent of what she has done.
On the other hand, all the children are obliged to be dutiful to their parents and it is impermissible for them to sever their relations with either of the parents, whether they remain married or get divorced.
For more benefit, please refer to Fatwa 87019 and 95504.
Allaah Knows best.