I think you are aiming to explore two overlapping problems. Preprocessing time-trajectories and clustering time-trajectories.
Functional data analysis (FDA) and in particular the methodology behind Multivariate Functional Principal Components seems like a potential avenue for what you want.
In respect to preprocessing:
Dynamic Time Warping  is just one of the multiple and earliest (prior to 1980) ways of exploring the curve alignment problem. Of the top of my head I can think of "pairwise synchronization" approaches, "quantile synchronisation" approaches, "self-modelling warping functions" approaches as well as "square-root velocity functions" approaches. All these approaches try to align data registered over a continuous domain (most often than not, time).
In respect to clustering:
Clustering trajectories has been a task visited by multiple scientific communities. I would not confine my search to Statistics community; for example, Zheng  gives an excellent overview of what takes place on the Computer Science side of things. That said and turning back to FDA-based approaches, Jacques & Preda  offer a relatively nice overview of the matter. In general, most approaches take the approach "align, get relevant representation (i.e. one that encapsulate the function-like nature of the data and does not treat the data as a simple multivariate sample) and then cluster". There is a smaller community with model-based approaches where alignment and clustering are done concurrently (e.g. ).
As mentioned it seems the data you are concerned with are multivariate (or more accurately bivariate) so probably approached related to multivariate functional principal components (e.g. ) would be helpful to get a relevant representation prior to clustering. (You may also find my answer here on :How to consider different samples in functional data clustering? useful.)
Notice that earlier approaches like the DTW, focused on time-registration and generally regarded the amplitude changes as "not so important". That was a reasonable assumption for their working environment (e.g. Speech processing, later on vehicle trajectories) where difference in phases (i.e. when the amplitude of the function changed) are usually more important that difference in amplitude (i.e. how much the amplitude of function changed). For example, when doing phoneme registration we do not really care how loud the speaker talks or when we doing air-traffc registration we do not care a lot about a plane's cruising altitude. In contrast, biological/medical applications (e.g. how fast does a particular animal reach peak weight, and what that peak weight is) people often care about both.
For something quick out of the box I would suggest looking at the CRAN has a separate view on Functional Data Analysis. There is a specific section of algorithms addressing the task of clustering functional data.
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