Why would I want to transform a Pose in Frame A using a Transform From B to C?

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I will give you an answer assuming that when you say "using transforms on vectors" you mean that you use the operations exposed by something like tf. If this is not the case, comment below and I'll try to adapt the answer.

From a mathematical point of view, a transform with frame_id B and child_id C "contains" the pose of frame C expressed in the coordinate system of B (there are also other interpretations, but this is my favorite). The pose is commonly represented by the rotation matrix bRc and the translation vector bTc. If you take any vector expressed in C (namely, cV) you can obtain its coordinates in the frame B as: bV = bRc*cV + bTc. If you now take a vector expressed in an arbitrary frame A and do the same, the result probably does not have any geometric meaning. However, from a pure "operational" point of view, the operation is well defined: just multiply a matrix by a vector and sum the result to another vector. This "operational" step (the linear algebra bit) is what is actually implemented in C++/python libraries, eg, in tf::Transform (the code is the implementation of Transform*Vector). Since it works on a more "abstract" level (it really does just matrix multiplication, without knowing what the quantities represent) it cannot prevent you from doing "nonsense operations". Ensuring that what the program does really makes sense, is something that is left to you, the user :)