Why wouldn't solvents be interchanged in isotonic solutions?

While studying osmosis, my book remarks that

Two solutions having same osmotic pressure at a given temperature are called isotonic solutions. When such solutions are separated by semipermeable membrane no osmosis occurs between them. For example, the osmotic pressure associated with the fluid inside the blood cell is equivalent to that of 0.9% (mass/volume) sodium chloride solution, called normal saline solution and it is safe to inject intravenously.

On the other hand, if we place the cells in a solution containing more than 0.9% (mass/volume) sodium chloride, water will flow out of the cells and they would shrink. Such a solution is called hypertonic. If the salt concentration is less than 0.9% (mass/volume), the solution is said to be hypotonic. In this case, water will flow into the cells if placed in this solution and they would swell.

But why would the osmosis simply stop? If I'm understanding osmosis correctly, then it is simply due to the fact that there is an SPM separating the two solutions, and hence the solvent (Or rather, the one with the size smaller than what the SPM allows, although I doubt it'd make a very big difference for a dilute solution) flows from the lower to higher concentration and eventually, equilibrium is attained.

In this case, even if you do inject somebody with a saline solution with $$0.9%$$ mass/volume sodium chloride solution, the SPM of the blood cell should still trade the solvent from the injection with the material inside (Since equilibrium does not translate to no flow, as confirmed by experiments with heavy and light hydrogen in the formation of NH3) it and the blood cell should still swell up. Obviously, that doesn't happen, though, or else I would have quite a different memory of getting a shot! Where did I go wrong in my reasoning?

Note: In the above, I'm assuming that the fluid inside the blood cells is something other than water. Please correct me if I'm wrong.

Edit: I realized the error. It says in the above paragraph itself that the fluid is water.

• So did you figure out the answer to your question? If so, could you please create an Answer below? – Stanislav Bashkyrtsev Mar 27 '16 at 14:57