Minimum number of moves required to transfer items from source bins to target bins?
The problem is NP-hard, by reduction from the partition problem.

Here is the reduction.  Consider an instance of the partition problem, namely, integer $x_1,\dots,x_n$.  Create $n$ source bins and 2 target bins, with $x_i$ items in the $i$th source bin.  Set the capacity on the target bins to $(x_1+\dots+x_n)/2$.  Then there is a solution to your movement problem that uses exactly $n$ moves, if and only if there is a solution to the partition problem.

We can also show your problem is strongly NP-hard, by reduction from 3-partition.

As a result, you should not expect to find any efficient algorithm that always gives the optimal answer and scales to large problems.  You might try looking for heuristics, approximation algorithms, or algorithms that could be exponential time in the worst case but hopefully are often fast in practice.  For instance, you could look into simulated annealing, integer linear programming, and approximation algorithms for bin packing.