According to Rav Yisrael Belsky (and based on the reasoning of Rav Yaakov Kaminetsky) toothpaste which contains non-kosher ingredients (such as animal-derived glycerin) is only kosher when the majority ingredients are inedible (which is typically no longer the case today). The Chicago Rabbinical Council (CRC) mentions that some authorities posit that since one would not eat or serve toothpaste, it does not need to be kosher. Seemingly accordingly, Rav Gedalia Schwartz of the CRC rules that toothpaste does not need to be kosher but should preferably not contain glycerin that might have been derived from animals. (The Orthodox Union, on the other hand, seems to consider the presence of glycerin to be more problematic.)
At the end of the day, even according to the most lenient view, is there any basis for the practice of giving toothpaste without a reliable hashgacha to infants and toddlers (whether with flouride or without) if they are not old enough to reliably spit it out?