What I read personal travel is still very limited. https://www.eac.int/travelling-within-the-eac
"East Africans can now drive private cars across borders and across the entire EAC region, free of charge, for visits no longer than seven (7) days. ... Anyone using a private car has to fill a form, obtainable at border crossing points or local Revenue Authority office, which grants permission for free access for a week."
And you only have 11 official border entries for cars between all members. At all.
Work:
https://www.eac.int/working-in-east-africa You are free to work unless in the public sector, but you'd still need to pass immigration procedures, and need to get and pay for a work permit within 15 days. Better than the usual wait for 14 days at the border, but then you wait for 3 weeks at the immigration office, and you'd need to pay the work permit fee (unless in Rwanda and Kenya).