Looking a this entirely from the point of view of an experienced amateur astronomer, I'm pretty sure you are describing a simple lunar occulation. The Moon moves eastwards w.r.t to the stars and so the eastern side of the lunar disk will cover up stars to the east of the Moon periodically. When the Moon is a waxing crescent as it is at the moment his will make stars disappear behind the part the Moons disk which is not illuminated. The length of an occultation, or the time before the star reappears on the western side of the Moons disk (as we look at it) will generally be less than a hour. Your description of 'fell with time about half an hour' doesn't entirely make sense but I could interpret that as the length of the occulation.
I'm sure that others on this site will offer you a different version of your observation but I can tell you without any element of doubt that the star would not be closer than the Moon. Anyone who tells you otherwise doesn't understand the physical nature of stars.
I would agree with ChrisTP if your observation as made entirely with the naked eye. Satellites look exactly like stars and I guess your description of 'falling' could be applied if the 'star' was moving from the Moon towards the horizon.