This is a very good question that has been answered particularly poorly in the past. The concept of karma in the early Buddhist texts does indeed imply persistence of an effect long after the condition is ceases - in our terms storage of a memory somewhere or other. And dependent arising forbids any effect to outlive the cessation of its conditions.
In my informal (as yet unpublished) writing on this I have referred to this problem as the problem of Action at a Temporal Distance. I discovered this problem independently, while study the writings of Prof Collett Cox on Sarvāstivāda Abhidharma texts. However I subsequently discovered that Nāgārjuna has also complained about this in his Mūlamadhyamakakārikā:
tiṣṭhaty ā pākakālāc cet karma tan nityatām iyāt /
niruddhaṃ cen niruddhaṃ sat kiṃ phalaṃ janayiṣyati // MMK 17.6 //
If the action remains until the time of maturation, then it would be eternal
If it ceases, being ceased, how does it produce a fruit? [My translation]
Nāgārjuna's response was to argue that all the components: agent, action, maturation, fruit, sufferer were all "like illusions" - ie. because of śūnyatā these entities have only relative and not ultimate existence. Karma only operates at the conventional level.
Now the Theravādins chose to respond to this one way: They proposed that each short lived citta that arose as a vipāka also became a kamma that was a condition for another identical citta. This introduced a number of other problems, chief amongst was how to account for moments when there was no apparent citta - such as deep sleep, experiences of cessation (nirodhasamāpatti), and (of course) death. To solve this they invented the bhavaṅgacitta to bridge the gap. The Theravādin view was only relevant in Sri Lanka and South East Asia.
The Vaibhāṣikas went about dealing the problem in an entirely different way. They reasoned that if a vipāka could be experienced in the future then the cause (hetu) must still exist as a condition in the future. And similarly for past karmas that we experience in the present. This earned them the nickname Sarvāstivādins because they believed that dharma always (sarva) exist (asti). The Sarvāstivādin view dominated North India and in China.
The Sautrāntikas opted for a similar approach in the form of a metaphor. Karma, they argued, was like a seed. The rice seed grows into a rice plant, but there is no direct connection from one to the other. This is an argument that karma is a "natural" process. Of course such metaphors are superficially pleasing, but explain nothing.
The Sautrāntika version was taken up by Vasubandhu and proposed as the solution to Action at a Temporal Distance in his Abhidharmakośa and it's autocommentary the Abhidharmakośabhāṣya (incidentally this is also the only extant source for the Sautrāntika view). And this became the Yogācāra view as well. Yogācāra exegetes did two things. They invented the alāyavijñāna as the repository of karma "seeds" and they reified the metaphor taking these images as facts. This version of karma overtook Nāgārjuna and became the standard Mahāyāna version.
Links above are to a series of essays I wrote on this problem that will also appear in my forthcoming book on Karma & rebirth.
Now unfortunately for us moderns, none of these proposed solutions really solves the problem of Action at a Temporal Distance. Karma cannot work with dependent arising. The ancients actually knew about this and we have their records of trying to fix the problem, but innovation simply stopped at some point and most of the theories died with the decline of Buddhism in India.
Discussion of this major problem in Buddhist doctrine simply ceased. The Theravādins retreated into formalism - an uncritical acceptance of whatever Buddhaghosa said and are leading the world in the production of apologetics for karma and rebirth. Similarly in the rest of the world the Mainstream became Mahāyāna and the Yogācāra view was taken on uncritically. The spirit of inquiry and problem solving simply went out of Buddhism and has yet to resurface in any major way. Regrettably not even academics seem to engage critically with Buddhist doctrine - they are in love with Buddhism it seems and reluctant to critique it. So while we get ever more refined views of history we see very little in the way of disagreement with the views in Buddhist texts - despite the fact that Buddhists of an earlier age are on record and thinking many of them demonstrably wrong.