(Abridged) The composition of the inner regions of protoplanetary disks is known to change with time due to the delivery of icy grains. Cold H₂O emission seen with JWST-MIRI is often hypothesized to be a tracer of this pebble drift. However, it is unclear to what extent processes such as photodissociation or the co-delivery of dust may impede such detections. We aim to obtain an improved, 2D view of transport in disks, to better understand how these processes can be traced by H₂O emission as seen with JWST-MIRI. We combine the 1D transport code DiscEvolution with the 2D thermochemical code DALI to create several grids of models in which the gas-phase abundances, dust properties, or both are varied according to the transport model. We consider scenarios with and without a traffic jam inside the H₂O snowline. The transport of both gas and dust leads to significant temperature changes within the disk, which strongly influence line fluxes and ratios. When a traffic jam is present, the delivery of H₂O can proceed unnoticed due to the co-delivery of dust. In addition, the relative strength of cold H₂O lines is not found to be sensitive to this delivery of gas-phase H₂O. Instead, the cold H₂O lines can be greatly enhanced by the delivery of only dust to the inner disk when a traffic jam is present, and one can create a spectrum with strong cold H₂O emission solely through the delivery of dust rather than H₂O. Recent work has used the 1500/6000 K H₂O line ratio as a proxy to determine the pebble mass flux crossing the H₂O snowline, but we find that this line ratio is rather sensitive to the temperature and dust distribution of the disk, and is often influenced not only by the cold H₂O mass, but also by the hot H₂O mass. This introduces complexities and trends as a function of time that do not match the true evolution of the pebble flux.