Is it permissible to perform the stoning on behalf of an incapacitated mother during Hajj, fearing for her safety due to the crowds, and is there any consequence for her or for the one who stoned on her behalf?

If a mother appoints her son to perform the Rami (stoning) merely out of fear of missing the bus, even though she is capable of performing it without hardship, or if she did not authorize him to do so, then she must offer a sacrifice (dam) for abandoning the Rami. However, if she has a valid excuse, such as illness or the like, then there is no harm in her appointing someone.