I will be in India and want to know if a non-Jew can get me around on Shabbat by rickshaw (a carriage drawn by a person on a bicycle or by foot) if I pay him before Shabbat? Answer:
Let us start with the bicycle-driven rickshaw and its smaller problems. When bicycles became popular, many poskim discussed their use on Shabbat, and almost all forbade it, for one or more of the following reasons. 1) Uvdin d’chol – This is a weekday-like activity, for amongst other reasons, it is a mode of transportation that takes people to many places for many purposes, including non-Shabbat-appropriate ones (see Tzitz Eliezer VII:30). 2) Bicycles often need repairs that the rider performs while forgetting about Shabbat (see ibid. and Yaskil Avdi III, Orach Chayim 12). 3) One might ride outside the techum Shabbat (boundaries of travel outside the city). While Rav Yosef Chayim of Bagdad (Rav Pe’alim III, OC 25, responding to the community of Bombay, India) dismissed these issues (some say he later changed his mind), the consensus of both Ashkenazi and Sephardi poskim and the broad minhag is to forbid it.
One might claim that when the vehicle is being ridden by a non-Jew for a Jewish passenger, one can be much more lenient, and a rickshaw thus operated might be too uncommon to have a clear minhag. While not totally dismissing this approach, the logic is somewhat difficult. The Jewish participant is taking part in the weekday-like activity, he could (help) fix the rickshaw if necessary, and he could be taken out of the techum Shabbat.
These problems do not seem to apply to a man-driven rickshaw, which is conceptually similar to a baby carriage or a wheelchair. While there is what to discuss in their regard (see Simchat Cohen, OC 78-79), the accepted consensus is clearly to permit them.
However, there is another problem in some places, including all of India. Pushing or pulling a carriage through the streets, by foot or by bicycle, is a form of carrying. Since there are not eiruvin in India, this is an additional problem on Shabbat. Admittedly, there are some mitigating factors. Carrying or pulling a person who is able to walk himself is forbidden only rabbinically because of a concept called "chai nosei et atzmo" (Shabbat 94a). While here the driver/puller is also transporting the rickshaw that they are sitting in, we say that the carriage is just an extension of the person (see Shabbat 93b). (The Jew cannot be carrying anything that he could not take into the streets himself.)
Another factor is that there is no Torah-level violation when the carriage is large enough to be its own "private domain" (Shabbat 8a – the details are beyond our present scope). While, even after these factors, there is still a rabbinic prohibition, one must remember that you asked about a non-Jew doing the work. In fact, Rav Yosef Chayim (in the aforementioned responsum) did permit a non-Jew pedaling a rickshaw that was of the right dimensions, but only if the Jew needed it to enable him to do a big mitzva. In general, one is allowed to ask a non-Jew to perform what is for us a rabbinic prohibition in order to facilitate a mitzva (Shulchan Aruch, OC 307:5). While we saw that many disagree because of the issues of bicycles, it is theoretically possible to use this leniency for a person-drawn rickshaw in the case of specific mitzva that one cannot get to in another way, which is unusual for a healthy person.
Paying a non-Jew in advance for that which he will do on Shabbat is not permitted when the non-Jew is required to do the work (that is forbidden for a Jew) specifically on Shabbat (ibid. 247:1). This is all the more so when it will be done with the active participation of a Jew. In this case, various forms of hinting will also not be effective.
The rabbinic community might want to discuss having non-Jews transport Jews in rickshaws on Shabbat for extreme cases like the mitzva or crucial needs of the sick or infirmed. However, under normal circumstances, this system is forbidden, all the more so in an area that does not have an eiruv.