The early Jerusalem church worked hard to relay what they knew of the historical Jesus to the daughter churches and subsequent generations.
Consider two reasons to agree:
If the Jerusalem church did really put effort into disseminating their Jesus-traditions to Christians throughout the Mediterranean (and the world), then it increases the probability that that they in fact did pass it down, i.e. that they accomplished what they set out to do.
Christians throughout the Mediterranean (Rome, Ephesus, Corinth, etc.), both Gentile and Jewish, regularly visited the apostolic church there in Jerusalem.
If Christians often visted the apostolic church, where the witnesses resided, it is only natural that they would fellowship with Christians there and be exposed to local teachings and traditions (especially from all the witnesses who attended). Not only would they desire to do this, but the Jerusalem church there would itself be eager to train and equip these visitors.