We study a dynamic boundary, e.g., a mobile impurity, coupled to N independent Tomonaga-Luttinger liquids (TLLs) each with interaction parameter K. We demonstrate that for N ≥ 2 there is a quantum phase transition at K ≥ 1=2, where the TLL phases lock together at the particle position, resulting in a nonzero transconductance equal to e2=Nh. The transition line terminates for strong coupling at K 1⁄4 1 − ð1=NÞ, consistent with results at large N. Another type of a dynamic boundary is a superconducting (or a Bose- Einstein condensate) grain coupled to N ≥ 2 TLLs; here the transition signals also the onset of a relevant Josephson coupling.