What, if anything, obligates a student to work with an advisor who lobbied for him/her to get funding? In my department, PhD admissions and funding are two separate processes, which means many PhD students are admitted even if there is no funding for them. To be admitted, an applicant typically needs to impress a faculty member enough to want him/her to take the applicant on as a student (which means this faculty member becomes the applicant's advisor, if s/he decides to accept the offer of admission). To get funding, an applicant need to impress at least one of the several faculty members who controls a training grant enough to want him/her to offer the applicant a slot on the grant. It is very uncommon (if not impossible) for any PhD student to receive funding outside of a training grant via an RA, TA, or other source, at least in the 1st year.

As such, it is fairly common for a professor who wants a particular student to join the department as his/her advisee to lobby for the student with one of the training grant directors. This is particularly true in cases where the student of interest has communicated that s/he has a funded offer from a rival department and will likely not consider our department without a similar funding offer being extended. 

Because of this practice, a difficult situation sometimes arises when funded students want to switch into a research area that is still within the scope of their training grant but not within the scope of their advisors' interests. Because both areas fall within the goals of their grant, these students are not risking their funding by switching (although they may be risking an additional year of PhD studies, depending on how late they switch). However, because their initial advisors usually had a large role in helping them secure funding, they are hesitant to make the switch out of a feeling of obligation to their advisor (e.g., "Dr. Z did a lot of work recruiting me and helping me get a funded offer, so I feel bad leaving him"). 

In situations where funding is not directly tied to one's advisor (but may be indirectly tied to him/her), what obligations does the student have to his/her advisor? Should the student feel free to switch advisors if s/he discovers that another research area appeals more to him/her? Are there any special considerations or etiquette protocols?
I think it is hard to answer this question without actually knowing the system and seeing whether such switches are commonly done. Typically, admission systems are either central (e.g., US-style, you apply with the faculty and once you are admitted you get an advisor) or decentral (e.g., central European, you apply and get funded directly by a specific professor). In the former case, transfers are not entirely uncommon. In the latter case, transfers are a bit of a dicey topic as students really only apply with and get accepted by one specific professor.

Your system sounds a bit like a mixture of both - somehow, admissions are centralized, but it sounds like individual professors have so much say in the process that it in practice may well work like a decentral system. One important question may be how this "lobbying" for students works in practice. Does every professor have an "alotted" number of students that they can choose (either explicitly or by common practice)? That is, does the transferring student "take away" an alotted place for a professor and add one for another? If that is the case, I would assume that transferring will indeed lead to some hard feelings. However, if the system is more flexible, and "lobbying" for a student really just means that the advisor sent a recommendation without many more implications, I think transferring may be just fine for everybody.

In the end, I think you need to see whether transfers happen regularly, and how the students that transferred are seen by the old and new faculty. If they never seem to happen, I would assume that professors will not be thrilled about the idea.