The objective, as outlined in the NPRM, for medical review of driver licenses is to determine if there are behavioral issues which may impact the judgment of a pilot. Drugs, alcohol, and other risk behaviors are screened and evaluated. There are cases where excessive speeding tickets and reckless driving has contributed to the non-issuance or revocation of a medical certificate.
To be clear, common traffic violations, are normally of no concern.
The FSDO is not involved in the medical process, but is involved in the enforcement process, so officially, they can be of little help. The Aeromedical Certification Branch of the FAA is the responsible branch. Substance dependence, substance abuse and personality disorders which involve overt acts are disqualifying.