I have heard opinions varying wildly on the subject and am trying to make sense of them halachically. Is it illegal on Shabbat to push someone in a wheelchair, let's say, from their home to synagogue outside of an eruv? If so, why? A friend of mine informed me that it is definitely permissible to do, since nothing is being carried.
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The English version of Sh'mirath Shabbath K'hilchathah (34:27) says that the disabled person in the wheelchair may propel himself through a rabbinic public domain and only for the purpose of a mitzva. The only person other than himself who would be permitted to push him would be a non-Jew. The allowance for the disabled person himself to do so is not due to the fact that moving the chair is not subject to the laws of carrying. It is. Although on the Torah level, the only way to incur liability for carrying through the public domain involves picking up and placing down the object, rabbinically all other forms of long (-er than 4 amos) range propulsion are prohibited.
More, very relevant details can be found here.
In the words of @msh210, CYLOR.
Yes, its hotza'ah (bringing out, carrying). Carrying does not literally mean in one's hands, easy examples of that are throwing things, and kicking things.