Transmodify works to send ancient gear knight and soldier to summon ancient gear engineer. However, it does not work to send ancient gear engineer to the graveyard to summon ancient gear beast, despite being the same type, attribute, and one level higher. Any reason for this?
Read the text of Ancient Gear Beast.
There's a reason why all Ancient gear special summon cards specifically state "ingore the summoning conditions". If that text is not there, you can't summon any high-level Ancient Gear effect monster.
Ancient gear beast clearly states that it cannot be special summoned