Abstract: Planet-disc interactions build up local pressure maxima that may halt the radial drift of protoplanetary dust, and pile it up in rings and crescents. ALMA observations of the HD 135344B disc revealed two rings in the thermal continuum stemming from similar to mm-sized dust. At higher frequencies the inner ring is brighter relative to the outer ring, which is also shaped as a crescent rather than a full ring. In near-IR scattered light images, the disc is modulated by a two-armed grand-design spiral originating inside the ALMA inner ring. Such structures may be induced by a massive companion evacuating the central cavity, and by a giant planet in the gap separating both rings, that channels the accretion of small dust and gas through its filamentary wakes while stopping the larger dust from crossing the gap. Here we present ALMA observations in the J = (2 – 1) CO isotopologue lines and in the adjacent continuum, with up to 12 km baselines. Angular resolutions of similar to 0 ''.03 reveal the tentative detection of a filament connecting both rings, and which coincides with a local discontinuity in the pitch angle of the IR spiral, proposed previously as the location of the protoplanet driving this spiral. Line diagnostics suggests that turbulence, or superposed velocity components, is particularly strong in the spirals. The (CO)-C-12(2-1) 3D rotation curve points at stellocentric accretion at radii within the inner dust ring, with a radial velocity of up to similar to 5 per cent +/- 0.5 per cent Keplerian, which corresponds to an excessively large accretion rate of similar to 2 x 10(-6) M circle dot yr(-1) if all of the CO layer follows the (CO)-C-12(2-1) kinematics. This suggests that only the surface layers of the disc are undergoing accretion, and that the line broadening is due to superposed laminar flows.