In California, the question is largely answered by EDC §44808, which says that
Notwithstanding any other provision of this code, no school district,
city or county board of education, county superintendent of schools,
or any officer or employee of such district or board shall be
responsible or in any way liable for the conduct or safety of any
pupil of the public schools at any time when such pupil is not on
school property, unless such district, board, or person has undertaken
to provide transportation for such pupil to and from the school
premises, has undertaken a school-sponsored activity off the premises
of such school, has otherwise specifically assumed such responsibility
or liability or has failed to exercise reasonable care under the
In the event of such a specific undertaking, the district, board, or
person shall be liable or responsible for the conduct or safety of any
pupil only while such pupil is or should be under the immediate and
direct supervision of an employee of such district or board.
There is an exception created by the courts. In Hoyem v. Manhattan Beach City Sch. Dist., the court decided that when "school authorities' negligent supervision of students on school premises, a pupil leaves the school grounds during school hours and is subsequently injured by a motorist", the school is liable. The negligence arises from allowing the child to leave without permission. Guerrero v. South Bay Union School District rebuffed a plaintiff attempt to assign liability when a child left with permission, and was injured. Plaintiff had argues that the school "failed to properly supervise the student in allowing him to leave", but California law does not allow schools to perpetually incarcerate students in order to avoid liability.
That said, in your scenario, the district has undertaken to transport the student to some drop-off site. So the question then is what duty the school districts has assumed to the student. In my non-California district, they may shoulder the burden of transporting a student from school to a designated drop-off point, and in the case of pre-school and kindergarten students the child can only be released to a designated adult, or else with a designated older sibling. In such circumstances, a school district can be argued to have assumed a responsibility to transport the child from school to a drop-off point, and were negligent in allowing the child to wander free-range. That would be a fact-intensive inquiry, since making a bus available does not per se create a duty to force the child onto the bus.