How to swabs
It is common practice to swab an instrument in the bell-to-mouthpiece direction. This has several disadvantages. Should a knot develop in the swab (and it commonly does), it will become stuck at the narrowest point, i.e. the register tube. There is a good chance that it will require a professional to remove it.
This picture shows what can happen when brute force is applied to try to get it out again. However, when pulled in the other direction, there will always be enough of the swab showing to remove it without force should it get stuck at the register tube.
Another reason for swabbing in this direction is that when pulled from the bell, the tone holes will slowly close on the top side of the hole as dirt is gradually pressed into them, causing possible intonation issues. When pulled from the other direction, any dirt will accumulate on the bottom side, which can make the tone a bit stuffy but won´t lower the pitch as much.
Cleaning the mouthpiece is essential but requires special care. As with the instrument itself, pulling through with a swab will cause gradual wear. It is better to use a soft cloth for the rails, table and chamber and an old-fashioned, woolen mouthpiece cleaner for the bore.