As KT said - anything you put in the eye - powder, liquid, ointment - is going to be flushed away in no less than 15 minutes, just with normal tear action; when tearing is in full flow - as we see with those pinkeye animals - it probably is washed out even faster.
I used to do the old subconjunctival injections of penicillin and steroids, back when I was in practice. No more. I don't think 'it keeps seeping out' for more than the few minutes it takes for that little needle puncture to seal over. Steroids inhibit healing of corneal ulcers... so I don't know how that came to be such an accepted practice... guess most of them healed up in spite of it, and folks made a faulty assessment.
If, however, you treat systemically, with an appropriate dosage of an appropriate drug - like the long-acting oxytetracycline formulations, Draxxin, etc., those drugs are present at therapeutic levels in the tears, so the eye is constantly being bathed with what should be an effective level of the drug - provided the organisms involved are susceptible to the antibiotic used.
That said, I usually keep a tube of mastitis treatment on hand and squirt some in the affected eye before I turn 'em out... knowing it's gonna be flushed out pretty quickly... but it's not the principal treatment.
I used to sew lids together or sew third eyelid to the upper lid. Nowadays, I'd just glue on a patch and call it good - I think the results are just as good.
As always, consult your own veterinarian.