Ajam, that is ethnic Iranians (in Bahrain also denoting shia Iranians), have lived along both sides of the Persian Gulf coast for centuries. The modern Ajam typically come from a merchant background, and it was typically commerce which would bring Iranians to the south-western shores of the Gulf. The Ajam of Bahrain for instance for a long time were considered distinctly from the native Shia Arabs. Apart from their shared faith, they would be different in many other ways - different trades, suburbs, and even political affiliations. These differences eroded over the course of the 20th century as more Shia Arabs engaged in business. The slow demise of Pan-Arabism as a political movement, and the rise of Islamist thought (particularly after the 1979 Iranian Revolution) led to a further erosion of these difference, as people identified as Shia first, rather than by Arab or Iranian first as they may have done in the past.

You may be asking why they did not 'return' to Iran? Most if not all Ajam will speak both Farsi and Arabic, so did not forget their heritage. And yet, families would have settled where their prosperity lay. Bahrain, Kuwait and Dubai are all important ports in the east-west route from India through Iran and the Middle East towards the Levant. For the same reason, there are ethnic Indian and non-Gulf Arab families which lived in Bahrain for many generations. While these groups are not necessarily Arabised, they have settled down in their respective places.