I haven't seen the promotion but the active ingredient in eyelash serums is a prostaglandin called bimatoprost which is used to treat high pressure inside the eye including glaucoma. It is now being promoted as a beauty product because patients who used it noticed a side effect, that is gave them longer, darker and fuller eyelashes. However, the side effects can be:
increased tear production
burning, dryness, itching, or redness of the eye
headache
darkening of the eyelid, eye color, or whites of the eyes
eye or eyelid discharge
allergic reactions: hives, rash, itching, chest tightness, swelling of the face, lips, tongue, or throat
vision changes
sensitivity to light
conjunctivitis
severe pain, swelling, irritation, or inflammation of the eye
severe eye fatigue
It shouldn't be used without consultation with a doctor if you have had:
eye surgery
swelling, inflammation, infection, or injury of an eye
lens replacement or missing eye lens
any other eye conditions
liver disease
kidney disease
I doubt I'll be risking any of those things trying to plump up thinning lashes.