Head Start Improvement Act of 2019 This bill replaces the existing Head Start program with block grants to states and Indian tribes for prekindergarten (pre-K) education. Instead of providing direct financial assistance to Head Start agencies, the Department of Health and Human Services (HHS) shall allot block grant funds for pre-K education among eligible states and Indian tribes in accordance with their relative proportions of children, age five and younger, from low-income households. Grant recipients shall use the grant funds to (1) award subgrants to eligible entities that provide pre-K education programs; (2) administer such programs; and (3) provide technical assistance, oversight, monitoring, research, and training. Under current law, HHS is authorized to designate, monitor, and establish standards for Head Start agencies. The bill instead shifts pre-K program oversight and control to states and Indian tribes, which shall have full flexibility to use grant funds to finance the pre-K programs of their choice. In addition, grant recipients may use grant funds to establish portable voucher systems that allow costs to be paid for attendance at private pre-K education programs. Under current law, federal financial assistance for a Head Start program is generally limited to 80% of total program costs. The bill maintains this limitation by requiring grant recipients to provide matching funds equal to 20% of the grant amount.