A tire assembly and method includes one or more elongate air passageway formed within a tire component, such as a tire sidewall. The air passageway is configured as a series or string of elongate cavities, adjacent cavities connected end to end by an elongate connecting channel. The connecting channel is dimensioned having a channel diametric size smaller than a cavity diametric size. Positioned within the tire component, the air passageway sequentially collapses segment by segment as each of the cavities pass sequentially over a rolling tire footprint. Air is pumped by the sequential air passageway collapse with the smaller dimensioned connecting channel(s) acting as valve components to directionally keep the pumped air moving between an air passageway air inlet and an air passageway air outlet and from there into the tire cavity.