With 16-month of Kepler data, 15 long-period (40-265 days) eclipsing binaries on highly eccentric orbits (minimum e between 0.5 and 0.85) are identified from their closely separated primary and secondary eclipses (∆t I, II = 3-10 days). These systems confirm the existence of a previously hinted binary population situated near a constant angular momentum track at P(1 - e 2)3/2 ~ 15 days, close to the tidal circularization period P circ. They may be presently migrating due to tidal dissipation and form a steady-state "flow" (~1% of stars) feeding the close-binary population (few % of stars). If so, future Kepler data releases will reveal a growing number (dozens) of systems at longer periods, following dN/dlgP vprop P 1/3 with increasing eccentricities reaching e → 0.98 for P → 1000 days. Radial-velocity follow-up of long-period eclipsing binaries with no secondary eclipses could offer a significantly larger sample. Orders of magnitude more (hundreds) may reveal their presence from periodic "eccentricity pulses," such as tidal ellipsoidal variations near pericenter passages. Several new few-day-long eccentricity-pulse candidates with long periods (P = 25-80 days) are reported.