Bismillahi Al-Rahmani Al-Rahim

Consecutiveness When Making Tayammum

The way you described the kayfiyyah of tayammum is confusing. How come in the first one (the obligatory acts of tayammum) you include wiping the hands/wrists but not in the description? And how come you included beating the ground in the description but not in the obligatory acts? What is the correct way of performing tayammum!?

You wrote ''consecutiveness in the minor ritual purity." as a condition for the validity of tayammum. What does consecutiveness mean here?

al-salamu `alaykum wa rahmatullahi wa barakatuhu

The text from the lessons is

(b9.3)  (The essential and obligatory actions)

It is obligatory (wajib) to make tayammum from pure, unburned, dirt.
Its obligatory (fard) actions are:
(a)     wiping the face;
(b)     wiping the hands up and including to the wrists;
(c)     being in order;
(d)     consecutiveness in the minor ritual purity.
It is a condition to intend the minor or major ritual impurity that tayammum is being made for. If he intended one it does not suffice for the other.
If he intended something supererogatory or did not qualify he does not pray an obligatory (prayer) with it. If he intended it (b: meaning: an obligatory) then he prays obligatory and supererogatory (prayers) during the entire time. (m: Basically: intending the obligatory lets you do the supererogatory but not the other way around.)
A nd:

(b9.6) (The Description of Tayammum)

Its description is:
(1)     intending;
(2)     then saying Bismillah
(3)     beating the hands on the ground with the fingers spread out;
(4)     wiping the face with the inner surface of the fingers and the back of the hands with the palms, and lacing fingers together.
I f you look closely, b9.6(4) describes how to accomplish both b9.3(a) and (b). The inner surface of the fingers are used for the face and the palms for the hands.

As for beating the ground, while the substance used (pure, unburned earth) is an obligatory part of making tayammum, how this is gotten is not. The description lists the most common situation (getting it from the ground), but you could also get it off of the wall or your pet cat. The correct way of performing tayammum is largely dependent on where you are. If you're on a road trip through the Cascade mountains in the middle of winter, you'll probably end up wiping your hands over your car and not striking or slapping them on the ground.

As for consecutiveness, here it is determined by making it analogous to when making wudu.

And Allah knows best.

wa al-salamu `alaykum