Tensor Decomposition in TensorFlow for multinomial time series dimensionality reduction
"Each time length can different, but there are the same features for each sample." You can make them all the same length, see Myth 1 of 
http://www.cs.ucr.edu/~eamonn/DTW_myths.pdf

"Want to end up with is a set of samples with a single feature in time" This might work, if the multinomial time series are "Dependent" See fig 1 of
http://www.cs.ucr.edu/~eamonn/Multi-Dimensional_DTW_Journal.pdf
Otherwise, this is a very very bad idea.

"With that, I can do clustering using distance determined by dynamic time warping." Sure, but you can do clustering with multinomial time series under DTW. See [a].

--

[a] Nurjahan Begum, Liudmila Ulanova, Jun Wang, Eamonn Keogh (2015). Accelerating Dynamic Time Warping Clustering with a Novel Admissible Pruning Strategy  SIGKDD 2015