You have several questions. Let's take them one by one :
What is teleported ?
In continuous variable (theoretical) papers, position and momentum are just convenient denominations for any pair of conjugate continuous variables, in the same way than a qubit can be a 2-level atom, a polarized single photon or a spin-1/2 particle. If a quantum object has continuous unbounded degree of freedom, it behaves like the position (along a given coordinate axis), and there exists a complementary observable which behave like momentum. Current experimental realizations include the quadrature of the light, the polarization of a bright beam, collective spin of atomic clouds, etc. Therefore, I'm not sure that focussing on the position of the particle itself helps in understanding these papers.
However, since, as told above, everything is equivalent to a position, the paper should apply to the position of a particle. So in the following, I will assume we are dealing with the position $x$ along the $x$ axis, and $p$ will be the momentum along the same axis.
What origin ?
The origin does not matter, as long as it is fixed. Suppose Alice and Bob have two different labs, and Alice wants to teleport the position and momentum of a particle from her lab to Bob's lab. That means that the initial position of particle A, relative to Alice's apparatus is the same as the final position of particle B relative to Bob's apparatus. In that sense, they can have the "same" position, even if they are in different rooms.
Of course, if one want to add details one should add that :
- Since we are in the quantum world, "the same position" means that every position measurement would lead to the same measurement statistics.
- Thank to Galilean relativity, if Alice and Bob are moving in respect to each other, the "same momentum" is also to be understood as relative to the movement of each lab.
- Actually, teleporting both position and momentum allows to 'completely' teleport this degree of freedom, and the statistics of any measurement (e.g Fock-state projection) will be the same.
Teleportation procedure
I tend to think that continuous variable quantum information is often easier to understand in the Heisenberg picture, where the observable operators behave almost like in classical physics
Initial state
Alice's initial particle's position and momentum are described by the operators $\hat x_a$ and $\hat p_a$. The entangled pair is described by $\hat x'_a, \hat p'_a$ and $\hat x'_b,\hat p'_b$.
For convenience, I'll define $\hat x_\pm=\frac1{\sqrt2}(\hat x'_a\pm x'_b)$ and $\hat p_\pm=\frac1{\sqrt2}(\hat p'_a\pm p'_b)$. The entanglement of the pair means that, initially $\hat x_-$ and $\hat p_+$ are both small. It is allowed
only if $\hat x_-$ and $\hat p_+$ both have big fluctuations.
The Bell measurement
The bell measurement is a simultaneous measure of $\hat x_m=\hat x_a-\hat x_a'$ and $\hat p_m=\hat p_a+\hat p_a'$. This measurement is possible because both observables commute. But after the measurement induces a back action on the complementary observables ($\hat x_a + \hat x_a'$ and $\hat p_a + \hat p_a'$), which become very noisy and cannot be measured later. In particular, the state of the particle A after the measurement does not contain information about the initial state anymore
The teleportation itself
Alice tells Bob the values of $x_m$ and $p_m$, and he moves/accelerates his particle accordingly particle. After this step, we have
$$ \hat x_b'^{\text{final}}=\hat x'_b + x_m = \hat x'_b + \hat x_a - \hat x'_a=\hat x_a-\sqrt2 \hat x_- $$
$$ \hat p_b'^{\text{final}}=\hat p'_b + \hat p_m = \hat p'_b + \hat p_a + \hat p'_a=p_a+\sqrt2 \hat p_+ $$
Since both $\hat x_-$ and $\hat p_+$ are small, the final observables $\hat x'_b$ and $\hat p'_b$ correspond to the initial observables $\hat x_a$ and $\hat p_a$. The position and momentum of the first particle have been teleported onto another particle.
So if the initial particle was moving ($\hat p_a\neq0$), then the "target" particle is indeed moving at the end of the process.
