It is clear that a TM can simulate a changing TM, so you only need to show the converse. Let me allow to use the "stay" movement in the changing Turing machine. It is easy to remove this assumption, but keeping it makes the following argument more intuitive.  

You can do the following: start from a TM $T$ with set of states $Q$ and tape alphabet $\Gamma$. Then, create a changing TM $T'$ with states $Q' = Q \cup (Q \times \Gamma \times \{L,R\})$ and tape alphabet $\Gamma' = \Gamma \cup \{ \gamma \}$. 

Intuitively, state $q \in Q$ of $T'$ represents state $q$ of $T$, while state $(q, a, m) \in Q \times \Gamma \times \{L,R\}$ of $T'$ represents the fact that we wish to write $a$ on the current tape cell, then transition to state $q$, and move in the direction specified by $m$.

Replace each transition $(q, a) \to (q', b, m)$ of $T$ with the following transitions in $T'$:

- $(q, a) \to ( (q', b, m), \gamma, S)$, and
- $( (q', b, m), \gamma) \to (q',b, m)$

Intuitively, this replaces the writing of a symbol $b$ on the tape with two operations: 1) we write $\gamma$ without moving the head, and 2) we overwrite $\gamma$ with $b$, move the head in the intended direction, and transition to the corresponding state of $T$.



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This is a tedious simulaton of a TM with a changing TM if the "stay movement" is not allowed.
Define $\Gamma' = \Gamma \cup \{\gamma_1, \gamma_2\}$ and $Q'= Q \cup (Q \times \Gamma \times \{L,R\} \times \Gamma')$. 
Replace each transition $(q, a) \to (q', b, m)$ of $T$ with the following transitions in $T'$:


- $(q, a) \to ( (q', b, m, \gamma_1), \gamma_1, R)$, 
- $( (q', b, m, \gamma_1), x) \to ( (q', b, m, x), \gamma_2, L)$
- $( (q', b, m, x), \gamma_1) \to ( (q', b, m, x), \gamma_2, R)$
- $ ( (q', b, m, x), \gamma_2) \to ( (q', b, m, \gamma_1), x, L)$
- $( (q', b, m, \gamma_1), \gamma_2) \to (q', b, m)$