For the Dashnak period, they had examples of mail sent abroad to the Hairenik [sometimes Harenik] Press in Boston, USA This was an important Armenian diaspora publisher - a Google search shows it publishing books and journals until the 1970s.
The Ashford collection contained three Hairenik Press items; two were offered as single items and illustrated in the catalogue and the third - shown below - was included in a group Lot and not illustrated. None of these three items has transit or receiver cancellations but it seems likely that they - along with other covers to the same address - travelled and thus became available to collectors in the West. They would have gone first to Batum and then to Constantinople. The British would have been involved for the period during which they occupied Batum [ See Footnote *]. Click on Image to Magnify.