Notice that if $$$s$$$ starts with a $$$1$$$ we must move the entire string $$$s$$$ to $$$t$$$ at some point. Also Notice that if we perform the operation, the total number of occurrences of $$$01$$$ and $$$10$$$ across both strings can only decrease by one. This gives us an upper bound on the answer being the number of occurrences of $$$01$$$ and $$$10$$$ in $$$s$$$ adding one to this if it starts with the character $$$1$$$.

Now the following construction uses the same number of moves as the upper bound (thus showing it is the minimum number of moves):

If $$$s$$$ begins with $$$1$$$ then select the entire string $$$s$$$ and move it to $$$t$$$. Then repeatedly find the first character in $$$s$$$ or $$$t$$$ which is not equal to the character before it (note under this construction such an index can only exist in one string at a time) and selected the suffix starting from this character and move it to the other string. During this construction some prefix of $$$s$$$ will contain $$$0$$$s and some prefix of $$$t$$$ will contain $$$0$$$s, so after each move the total number of $$$01$$$ and $$$10$$$ will decrease.