it's a temporal thing to comment motion... physics (its change of meaning), dynamics (its change of meaning!), alchemy and chemical "reactions", political "movements", "revolutions"...
the drawing of the picture is good but the words untrue; the meaning of the cases is not motion, and prepositions are their modifiers in space and sense: they don't add motion. the sense of motion is only given by the phrase and meaning—most times by the verb first—, so, 'ab urbe condita', 'exuere armis', 'de manibus effūgit', 'ex innocentia nascitur dignitas', 'ad Caesarem venērunt'. when prepositions are particles in verbs and other words they show the same: they work as facing, local, spatial or temporal modifiers
ex doesn't have to mean 'from within', it is just the contrary peer of in, that doesn't have to mean 'in', as you know; see the phrase 'out of love'. de doesn't have to mean at all 'down from'; and ab doesn't have to mean that, is not the only one meaning that, and preposition is not needed to mean that, see the examples above
prepositions don't "govern" names; prepositions—poor name for words that may be also adverbs and conjunctions—are with cases by custom and mood: see the sundry uses in hellenic (varying in dialects and time) and old english
the accusative and ablative are spatial (time also) and real (res 'thing'). accusative to, for, through, throughout; and ablative from, by, out, of. the spatiality made them likely for prepositions and these fit well with the reality. the accusative is some that some faces or addresses ('o tempora'), mode ('acutum cernere'; and see the particle -im), quantity (seen as adverb, 'multum pluit', 'tertium consul'), length, heigth, breadth; so as you can see the meaning is not motional; and most times it is not true to say it is locational
the dative and genitive are real, essential (essentia 'being'), habitual (habitus 'property, feature'), belonging. dative to, for; genitive of. show: 'John's (g.) six-wheels (g.) car is given to Laura (d.) for her need (d.)'. the 'habits' of car are all the genitives and datives and we see the change of belonging (or 'habit') from John to Laura
more, 'the city of Liverpool (g.)', 'in the name of justice (g.)', 'hoc est mihi argumento (d.)'. city is had by Liverpool, name by justice; some receives as argument some. otherwise the meaning is of refering: 'what city? Liverpool' 'what name? justice'. in the other side: 'they gave city to Liverpool (d.)', 'he gave it (d.) justice as name (d.)', 'sunt nobis mitia poma', 'homini cum deo similitudo est', 'mihi nomen est Marcus'
so we see the two pairs meaning two fields of things
and we see that the two pairs accusative-dative and ablative-genitive mean sundrily 'end' and 'outset, start, beginning, origin'. in the ablative distance, absence, temporal, real or local cause, so the idea of spatiality (time included). the genitive means 'habit', 'real' origin or reality, no spatiality here. 'simulacrum ex auro' (likeness out of gold), 'simulacrum auri' (likeness of gold; or, gold's likeness)