  We investigate the enrichment in elements produced by the slow
neutron-capture process ($s$-process) in the globular clusters M4 (NGC 6121)
and M22 (NGC 6656). Stars in M4 have homogeneous abundances of Fe and
neutron-capture elements, but the entire cluster is enhanced in $s$-process
elements (Sr, Y, Ba, Pb) relative to other clusters with a similar metallicity.
In M22, two stellar groups exhibit different abundances of Fe and $s$-process
elements. By subtracting the mean abundances of $s$-poor from $s$-rich stars,
we derive $s$-process residuals or empirical $s$-process distributions for M4
and M22. We find that the $s$-process distribution in M22 is more weighted
toward the heavy $s$-peak (Ba, La, Ce) and Pb than M4, which has been enriched
mostly with light $s$-peak elements (Sr, Y, Zr). We construct simple chemical
evolution models using yields from massive star models that include rotation,
which dramatically increases $s$-process production at low metallicity. We show
that our massive star models with rotation rates of up to 50\% of the critical
(break-up) velocity and changes to the preferred
$^{17}$O($\alpha$,$\gamma$)$^{21}$Ne rate produce insufficient heavy
$s$-elements and Pb to match the empirical distributions. For models that
incorporate AGB yields, we find that intermediate-mass yields (with a $^{22}$Ne
neutron source) alone do not reproduce the light-to-heavy $s$-element ratios
for M4 and M22, and that a small contribution from models with a $^{13}$C
pocket is required. With our assumption that $^{13}$C pockets form for initial
masses below a transition range between 3.0 and 3.5 M$_\odot$, we match the
light-to-heavy s-element ratio in the s-process residual of M22 and predict a
minimum enrichment timescale of between 240 and 360 Myr. Our predicted value is
consistent with the 300 Myr upper limit age difference between the two groups
derived from isochrone fitting.
