  We investigate barium (Ba) abundances in blue straggler stars (BSSs) in two
open clusters, NGC 7789 (1.6 Gyr) and M67 (4 Gyr), as signatures of
asymptotic-giant-branch (AGB) mass transfer. We combine our findings with
previous Ba abundance analyses in NGC 6819 (2.5 Gyr) and NGC 188 (7 Gyr). Out
of 35 BSSs studied in NGC 7789, NGC 6819, and M67, 15 (43$\pm$11%) are
Ba-enriched; no BSSs in NGC 188 are Ba-enriched. The Ba abundances of enriched
BSSs show an anticorrelation with cluster age, ranging from an enrichment of
[Ba/Fe]$\sim$+1.5 dex in NGC 7789 to [Ba/Fe]$\sim$+1.0 dex in M67. The
Ba-enriched BSSs all lie in the same region of the HR diagram, irrespective of
cluster age or distance from the main-sequence turnoff. Our data suggest a link
between AGB donor mass and mass-transfer efficiency in BSSs, in that less
massive AGB donors tend to undergo more conservative mass transfer. We find
that 40$\pm$16% of the Ba-enriched BSSs are in longer-period spectroscopic
binaries with orbital periods less than 5000 days. Those Ba-enriched BSSs that
do not exhibit radial-velocity variability suggest AGB mass-transfer in wide
binaries by either wind mass transfer or wind Roche-lobe overflow. Given the
preponderance of long orbital periods in the BSSs of M67 and NGC 188 and the
frequency of Ba enrichment in NGC 7789, NGC 6819, and M67, it may be that AGB
mass transfer is the dominant mechanism of BSS formation in open clusters older
than 1 Gyr.
