Well there is no doubt that the body slows metabolic rate when you reduce calories or lose weight/fat. There are at least two mechanisms for this.
One is simply the loss in body mass. A smaller body burns fewer calories at rest and during activity. There's not much you can do about that except maybe wear a weighted vest to offset the weight loss, this would help you burn more calories during activity.
However, there's an additional effect sometimes referred to as the adaptive component of metabolic rate. Roughly, that means that your metabolic rate has dropped more than predicted by the change in weight.
So if the change in body mass predicts a drop in metabolic rate of 100 calories and the measured drop is 150 calories, the extra 50 is the adaptive component. The mechanisms behind the drop are complex involving changes in leptin, thyroid, insulin and nervous system output (this system is discussed to some degree in all of my books except my first one).
In general, it's true that metabolic rate tends to drop more with more excessive caloric deficits (and this is true whether the effect is from eating less or exercising more); as well, people vary in how hard or fast their bodies shut down. Women's bodies tend to shut down harder and faster.
But here's the thing: in no study I've ever seen has the drop in metabolic rate been sufficient to completely offset the caloric deficit. That is, say that cutting your calories by 50% per day leads to a reduction in the metabolic rate of 10%. Starvation mode you say. Well, yes. But you still have a 40% daily deficit.