My units can't move to a nme production unit without causing war.
You can do this, it is a very useful trick. I call it the "Diplosneak". Move your Diplomat onto a square your rival is eating. Then the NME will evacuate off the square. Next, you move a more permanent and defensively stout unit onto that square, and you can pull back your defenseless Diplomat. Now you own it, and the rival can only take it back by starting a war.
Many ways to use the Diplosneak trick. You may wish to get your offensive heavy-hitters onto a mountain adjacent to an NME city, from which they can attack more effectively. If you plan on bribing an NME city away, it becomes much more affordable if you can force it into civil unrest. To do this, use Diplosneak to occupy the high-food squares around the rival city. Soon, they can't support themselves, they go into turmoil, and you can bribe the place for pennies (and not start a larger war).
Sometimes i want to put a rival city in my pocket for later. In this case, i do not want them to have access to shields that they can use for units to break my encircling siege. Diplosneaking units onto that city's high-shield squares keeps the place from putting out dangerous units.
Even odder, sometimes i want a captive city to build Settlers which i can bribe away, since they'll become units belonging to NONE as their home city, thus they don't cost my civ any food or shields to maintain. In this case, i Diplosneak my own Settler onto the rival city's grasslands, lay down irrigation, then retreat my Setty, and let the rival take over the newly developed squares. With access to good food, the city starts putting out its own Settlers, as fast as i can bribe them away. It's like turning a rival city into a cow, and milking it.
If you ever want the rival to start a war so you can punish them, you can do the "Tethered Goat" trick. Identify the rival's most valuable square, a deer-forest or coal-hills, maybe a horse-plains or a shield-river. Then Diplosneak one of your weak units onto that square, and it will act like a grain of sand in an oyster. It irritates the rival so much that they will feel obliged to attack you.
