With the exception of Kayna's Vessel (which is always available), Vessels of different Monsters are available at different times. Unless all the necessary eggs are Zapped in the Vessel within the time allotted, the eggs will go bad and the Vessel will expire. However, you will receive compensation for the expired Vessel in the form of Relics. If the Vessel's inventory has been completed, you can transfer its egg to the Nursery at any time.