You are offline, so I'll leave my thoughts.
There are many causes for recurrent vomiting, but since both of your children have this symptom, an infectious cause (e.g., an intestinal virus) seems likely. Paleness around the lips would make me worry about dehydration, especially if either child does not urinate well at least once every 8 hours or if the inside of the cheeks is dry and sticky instead of moist and slick--if you note either of those symptoms (decreased urination, dry mouth lining), the inability to hold down even an ounce of liquid, or blood (bright red, maroon, or black) or bile (light to dark green) in the vomit, then a prompt ER visit is advised. Other data follows:
Prompt evaluation by the child's physician or in an emergency room is advised if you note any of the following in association with persistent vomiting/diarrhea: fever that is high or lasts beyond 48 hours; blood (bright red, maroon, or black) or bile (light to dark green) in the vomit; blood (bright red or tarry black) or mucus (copious or bloody) in the bowel movements; diarrhea that lasts beyond 7 days; failure to urinate well at least once every eight hours; marked dryness or stickiness (not slick moistness) when a clean finger is swiped against the inner cheek; inability to take or hold down even one-half teaspoon sips of liquid; markedly low energy or responsiveness. If none of the above urgent criteria are met, then the cause of the vomiting/diarrhea may be viral, especially if there has been exposure to sick family or friends with similar symptoms. Please note that antibiotics do NOT work against viruses. I would also avoid anti-diarrheal meds unless advised by the child's physician (in a sense, these prevent the body from ridding itself of the viral infection). You may use Tylenol for ache or fever, but avoid ibuprofen (Motrin or Advil) as it may irritate the stomach.
To prevent dehydration and maintain nutrition, try this routine called "2 bites, 2 ounces."
Alternate every 10-20 minutes between offering:
If age 4-11 months:
If age 12 months and up
* Start with a measuring teaspoon--once the child has tolerated this a couple of times, go up to two teaspoons or a measuring Tablespoon. A measuring Tablespoon used for baking holds ½ ounce of fluid, two Tablespoons equals 1 ounce of fluid, and a measuring cup will help when giving larger volumes.
** In place of Pedialyte and similar over-the-counter oral rehydration solutions, you may use the following mixture of 1 level measuring teaspoon of salt and 8 level measuring teaspoons of sugar stirred until dissolved in 5 measuring cups (34-40 ounces) of clean drinking water.
*** At age 8 months and up, you may give plain yogurt labeled as having "active cultures" or "Lactobacillus cultures." You may sprinkle some sugar to improve flavor. Avoid yogurt with added fruit.
If the child refuses solids, offer the liquids instead, gradually increasing the fluid volume up to 4 ounces as tolerated (stay at the fluid volume where no vomiting occurs). If the patient manages enough fluid intake to urinate at least once every 8 hours, this will help to prevent dehydration. Starchy solids promote formed stools. Eating plain yogurt helps to restore normal intestinal function, while a more effective over-the-counter option similar to yogurt is 1 capsule daily of "Culturelle for Kids" mixed into a snack of cool or room temperature applesauce, mashed bananas, or plain yogurt.
Avoid liquids and foods that may promote diarrhea (examples: prunes and pears; prune, pear, apple, or red grape juices; unpeeled fruits; bran- or whole-grain type cereal, bread, waffles, or muffins).
Viral sources of vomiting/diarrhea often resolve within 7-10 days. Focus on hydration, and remember the urgent criteria at the start of this message.