There are several things that I have to keep in mind when it comes time to move seedlings. Foremost is to be careful -- large hands and clumsy digitS'.
There is no choice about disturbing the roots. Through photosynthesis, damage is likely to be repaired. Photosynthesis is mostly taking place in the leaves. The seedlings with healthy leaves can recover from transplanting shock.
Allow the plants to develop their true leaves. At one time, I might have realized the importance of leaves and thought that meant that I should grab them by the stems. Not for me! A Seedling has only one stem. If it is damaged, I've killed it. There is little choice but to hold them by the leaves. I absolutely cannot be trusted with seedlings as tiny as snapdragons. They are in DW's care. Other seedlings like tomatoes and peppers, I can handle but I want to see those true leaves even if I am mostly trying to hold them by the seed leaves.
Temperatures and sunlight should be in balance. Without instruments to measure, it's difficult to know how much light your window is admitting, difficult to know how much is being emitted by lamps. In photosynthesis, heat isn't a replacement for light. Plants aren't warm-blooded mammals although they metabolize nutrients within a range of temperatures. Growth occurs. Adequate warmth with too little light results in plants wandering off, looking for more light. We call that legginess.
Steve