There was also a school at Easton, a little bit further away than Kirkandrew Moat, but slightly closer to Netherby Hall.
In 1861 there was no legal requirement for children to attend school, so an education may have been a little hit and miss and dependant upon whether there was a school nearby, if it was free, or if not whether their family could afford the few pennies that it may have cost and whether the children were working. Some children's only education may have been via Sunday school or no education at all.
Its possible that your G Grandfather may have had an intermittent education as the family moved about, depending on what was available and no doubt, it wouldn't have been that long before he was working himself, possible with the "gang".