Given 2 sets of n points: minimize sum of traveled distances
As mentioned in the problem statement, this is the Assignment Problem (minimum weight bipartite matching) where it is known that the weights are the Euclidean distances. 

There have been several improvements since the Hungarian Algorithm, at least in terms of asymptotic bounds.  Depending on the exact size of the graph, any of several algorithms may be the best.  A table in the paper by Cohen, et al gives details.  Edmonds and Karp's algorithm is $O(nm + n^2 log n)$, and is still the best bound that doesn't depend on the maximum weight in the graph.  Cohen's algorithm appears to be the best for sparse graphs, which is not your situation.  I think the best for your dense graph would be Sankowski's $\tilde{O} (W n^\omega)$, since it doesn't depend on $m$.

I do not know if there are ways to exploit this problem's specific weight structure (Euclidean distances) for further improvements.

Sources:

Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in $\tilde{O}(m^{(10/7)} log W)$ Time.
Michael B. Cohen, Aleksander Madry, Piotr Sankowski, Adrian Vladu
https://arxiv.org/abs/1605.01717v3 (preprint)

J. Edmonds and R.M. Karp. Theoretical Improvements in Algorithmic Efficiency for Network Flow
Problems. J. ACM, 19(2):248–264, 1972.

Piotr Sankowski. Automata, Languages and Programming: 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, chapter Weighted Bipartite Matching in Matrix Multiplication Time, pages 274–285. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.
http://link.springer.com/chapter/10.1007%2F11786986_25