  By using the GRAVITY instrument with the near-infrared (NIR) Very Large
Telescope Interferometer (VLTI), the structure of the broad (emission-)line
region (BLR) in active galactic nuclei (AGNs) can be spatially resolved,
allowing the central black hole (BH) mass to be determined. This work reports
new NIR VLTI/GRAVITY interferometric spectra for four type 1 AGNs (Mrk 509, PDS
456, Mrk 1239, and IC 4329A) with resolved broad-line emission. Dynamical
modelling of interferometric data constrains the BLR radius and central BH mass
measurements for our targets and reveals outflow-dominated BLRs for Mrk 509 and
PDS 456. We present an updated radius-luminosity (R-L) relation independent of
that derived with reverberation mapping (RM) measurements using all the
GRAVITY-observed AGNs. We find our R-L relation to be largely consistent with
that derived from RM measurements except at high luminosity, where BLR radii
seem to be smaller than predicted. This is consistent with RM-based claims that
high Eddington ratio AGNs show consistently smaller BLR sizes. The BH masses of
our targets are also consistent with the standard $M_\mathrm{BH}$-$\sigma_*$
relation. Model-independent photocentre fitting shows spatial offsets between
the hot dust continuum and the BLR photocentres (ranging from $\sim$17 $\mu$as
to 140 $\mu$as) that are generally perpendicular to the alignment of the red-
and blueshifted BLR photocentres. These offsets are found to be related to the
AGN luminosity and could be caused by asymmetric K-band emission of the hot
dust, shifting the dust photocentre. We discuss various possible scenarios that
can explain this phenomenon.
