The transmission of Siddham eastward rests on a small number of long journeys. Chinese monks travelled to India to obtain texts, to study, and to settle questions that the translations available to them left open, and several of them wrote accounts of what they found. Xuanzang, in the seventh century, and Yijing, a generation later, are the best known, and both returned with manuscripts and with a first-hand knowledge of the language those manuscripts were written in.
Traffic ran the other way as well. Indian and Central Asian masters travelled to China and worked there, and in the eighth century three of them, Shubhakarasimha, Vajrabodhi and Amoghavajra, established the esoteric transmission at the Tang capital that Kukai later received. The texts on which Shingon rests were translated in that setting, by people for whom the Sanskrit original was not an abstraction.
What those journeys carried was not only documents but competence. A script cannot be transmitted by a manuscript alone, because a manuscript does not say how its letters are pronounced or how its conjuncts are resolved. It has to be taught by somebody who knows, and the chain of teaching that reaches Japanese Siddham runs back through these travellers to India. The letters survived in Japan because at each link somebody was there to explain them.