This is not a full answer, since I do not know the counterexample Lin refers tooto, but I can offer some explanations and guesses which are too long for a comment:
You can define a first variation for currents similarly to that for varifolds by considering $\frac{d}{ds}|_{s=0} E((h_s)_\# T)$ for some smooth enough family of diffeomorphisms $h_s$ with $h_0 = id$$h_0 = \operatorname{id}$ (and $h_s = id$$h_s = \operatorname{id}$ outside a compact set). Here $E$ is the energy you want to consider (i.e. mass in the case of minimal surfaces) and $h_\#$ denotes the pushforward. This should coincide with $|T|$$\lvert T\rvert$ being a stationary varifold though.
Regarding the boundary, you can fix it, by setting $\partial T = R$ and correspondingly only allowing variations with $h_s = id$$h_s = \operatorname{id}$ on $\operatorname{supp} R$. Otherwise, if you are not additionally penalizing the mass of the boundary, $\partial T=0$ is the only one that makes sense in the context of minimal surfaces.
I don't know the counterexample Lin refers to, but I think it would need to involve cancellation of parts with opposite orientation (otherwise $|\lim_{k\to\infty} T_k| = \lim_{k \to \infty} |T_k|$$\lvert\lim_{k\to\infty} T_k\rvert = \lim_{k \to \infty} \lvert T_k\rvert$ which is known to be stationary). Additionally I think it should not be a sequence of minimal currents, otherwise the limit will be minimal and thus stationary again.
