The gemara concludes that טלטול מן הצד is considered normal moving of muktzah and cannot be done on shabbos. Tosafos explains that there are other sources that imply that one can move muktzah through tiltul min hatzad, therefore Tosafos concludes that it depends on the purpose. If one is doing טלטול מן הצד for the sake of moving the actual muktzah item from one place to another such as by a corpse, that is not allowed. But if one is doing the טלטול מן הצד for the sake of a non-muktzah item, just that the muktzah gets pulled along with it, that is permissible. This distinction is paskened in Shulchan Aruch. In siman 311:1 the Shulchan Aruch writes that tiltul min hatzad for the sake of the muktzah is prohibited (unless it is a special circumstance for kavod ha'meis), but in 311:8 the Shulchan Aruch paskens that tiltul min hatzad for the sake of something permitted where the muktzah will inadvertently be moved, is permitted. Based on this, the Mishna Berura (6) writes that one may not take a non-muktzah item and use it to push muktzah because that would be טלטול מן הצד לצורך דבר האסור, which would be a violation of moving muktzah. However, if one is doing so because they need to use the place where the muktzah is sitting, it is permitted. Therefore, the M.B. (308:115) paskens that if one has shells on a table (assuming that it doesn't make the area look disgusting so that there is no heter based on גרף של רעי), since they are muktzah, he cannot move them with his hand. But one can take a napkin and sweep them away if he needs to use the place since that would qualify as tiltul min hatzad for the sake of something permitted.
The gemara paskens like R. Yehuda that if a fire breaks out in a house where there is a corpse, one may move the corpse away from the fire because chazal were concerned that if they didn't allow it, one would come to intentionally (Tosafos) extinguish the fire. The Ran cites a machlokes whether we only allow the moving of a corpse when it is טלטול מן הצד, even though it is normally forbidden, we allow it to be done to prevent extinguishing a fire, or do we even allow the direct moving of a corpse to save it from a fire. It seems from Tosafos that when it is not possible to move the corpse through a ככר או תינוק, or move it מן הצד, we allow the direct moving of the corpse. Tosafos writes that the fact that the gemara suggests that everyone holds that טלטול מן הצד is a problem, when it could have said the opposite, implies that we pasken that טלטול מן הצד (at least for the purpose of moving muktzah) is prohibited. Tosafos suggests that it could have said the opposite and made the אוקמתא to be when there is not ככר או תינוק and no beds so that it can't be moved with טלטול מן הצד. It seems that even if טלטול מן הצד wasn't possible, Tosafos assumes that R. Yehuda would still permit the direct moving of a corpse to save it from a fire. This is exactly what the Shulchan Aruch 311:1 paskens. Ideally, a corpse should be moved using a ככר או תינוק, if that's not possible then מן הצד and if that's not possible then it can be moved directly.