Having headed my own consulting firm for twenty years before recently going in house, it is clear every situation is different and certainly so when you are coming from having been an internal employee. Situations like this occasionally come up with disgruntled former clients and, although rare, do happen to the best of us. It's called scapegoating.
Smearing a former employee under any circumstance, whether to internal audiences or even more so to external stakeholders, is extraordinarily bad form on the part of the employer. Such conduct, under the right circumstances, may even rise to legal exposure, i.e., defamation. Especially if the allegations are not true, widely known to not be true, cannot be substantiated, and can be proven to negatively impact personal reputation and future employability (i.e., damages).
I favor the swift and direct response, especially if the former employee has a demonstrated track record of a consistent quality performance on that job. Try to get evidence or attribution on specific instances of the 'public' smear from reliable former colleagues that may be sourced blindly. Then have your attorney serve the former employer notice. In most cases this will terminate such behavior quickly. Many states do not look upon this type of passive-aggressive employer retribution kindly. And the courts are quickly evolving their view around employee rights - i.e., non competes and other matters.