Pare down would be a good choice.
1.2 Reduce (something) in size, extent, or quantity in a number of small successive stages.
‘union leaders publicly pared down their demands’
‘When he wrote the play, his intention was that there should be no
excess and so each scene is pared down to exactly what he wants to
used with list:
Advisers first help a student select 20 to 25 colleges, prodding the
student along until he or she pares down the list to the eight or so
to be considered seriously. (Guidance For Sale, TAMALA M. EDWARDS,
Time Magazine: 1999/11/01)
Oregon plans to expand its Medicaid program to cover all families
under the poverty line, but also to ration their care by paring down
the list of medical procedures Medicaid would pay for. (new game-plans
take shape in Washington, Christian Science Monitor , 1991/11/26)