After the four-day strike, Border Force staff will then work to rule and refuse to work overtime from 4 September until 22 September.
The union has said more than one in four members have taken time off due to stress or related mental health reasons since the introduction of the new rosters in April.
Most staff regularly worry about managing home commitments around work, according to the union.
General secretary Fran Heathcote said: “This survey shows how badly our members’ wellbeing is being affected by the new roster system.
“These inflexible rosters are preventing parents dropping off and picking up their children from school.
“Managers could easily resolve the issue by allowing greater flexibility and by agreeing fixed shifts that give our members the chance to plan their caring responsibilities.”
Border Force is responsible, on behalf of the Home Office, for passport control checks at all UK airports – including Heathrow.
A Home Office spokesperson said: “We appreciate the tireless work that Border Force do to keep our borders safe and secure, and we are committed to continuing our conversations with the union so we can find an agreement that works for both the public and staff.
“We will have robust plans in place to minimise disruption where possible, but we urge passengers to check the latest advice from operators before they travel.”