Holding a multichamber ocarina
Holding a multichamber ocarina is much the same as holding a single chamber. The right hand comes in from the rear with the mouthpiece facing toward you. The left hand comes in from the opposite side. A double is shown here, but the ideas are the same for ocarinas with more chambers.

As their name implies, multichamber ocarinas have multiple chambers, each of which has its own windway, voicing, and finger holes. The chambers of a quad are labelled in the next diagram. The holes within the area marked '1' are connected to the first chamber; those shown in '2' are for the second chamber, and so on. Chambers are additive, so if you are playing a double for example, 3 and 4 can be ignored.

Notice that the left hand is responsible for one row of holes, the same as on a single chamber. The right hand is responsible for two or more banks of holes. Normally, only one chamber is blown at a time and all right hand fingers move to the holes of the blown chamber. The exact fingerings of the higher chambers does vary between ocarinas, and some of these common variants are covered on the page The fingering systems of multichamber ocarinas. Checking a fingering chart for your instrument is recommended though.
The positioning of the left hand on a multichamber ocarina is identical to that of a single chamber: it only has to cover a single set of holes. As with singles, you should keep your fingers gently curved along their length. Joints with sharp angles usually indicate a poorly placed finger or poor hand posture. See 'How to hold an ocarina' for more.

As it is difficult to see what your fingers are doing in normal playing position, I recommend using a mirror. You can also bend your elbows to move the ocarina away from your mouth without influencing finger placement.
The right hand's position is also similar to on a single chambered ocarina, but it has two or three rows of holes to cover. While this looks complicated, it isn't in practice. When you playing higher notes, you move the fingers to the second chamber. Normally, only one of these rows is covered at any time.
The right thumb needs to be positioned to allow the fingers to cover their holes. If the ocarina has a right thumb hole, this will guide the placement of the fingers. Otherwise, the thumb should be positioned approximately in the middle of the two rows of holes above. On a triple, the thumb should rest roughly opposite the finger holes of the second chamber.
If you can bend your thumbs backwards, don't do so. The right thumb carries most of the weight of the ocarina; if it is constantly forced back, it may become painful. I have added a ramp to the ocarinas that I make, which allows the thumb to better support the instrument and stops it from bending backwards.

The action of moving the fingers between the chambers comes partly from the wrist and partly from bending the fingers. The thumb normally remains stationary. Please see the video below..