There are many satisfying answers, but not a single satisfying answer. I'll just drop a bunch of relevant literature without much explanation, and without much specific context; I'll maybe come back later for adjustments. Better than nothing, I hope!

• https://arxiv.org/abs/1904.08541
• https://arxiv.org/abs/1805.04346 (this is in a very precise sense the sharpest monad-theory correspondence one can get, by construction; yet, the notion of "theory" is quite general: no assumption on $\cal A$.)
• https://arxiv.org/abs/1112.3076
• https://arxiv.org/abs/0907.2460 Example 4.17, and below that for the multisorted version
• https://arxiv.org/abs/1511.02920 (if you ask me, this is the sharpest one can go while keeping at least some of the essential features of what an "algebraic theory" shall be)
• https://arxiv.org/abs/1307.2963 (if you ask me, this is a brilliant characterisation of Lawvere theories as categories enriched over $[{\sf Fin,Set}]$).
• https://arxiv.org/abs/1507.08710 see section 5, and below, where the authors a neat conceptual criterion for when an algebraic theory is the tensor product of two given ones.

The implicit starting point of all these different development deserves at least a word of explanation.

The idea is more or less the following: let's say that a "Lawvere theory" is an identity-on-object functor ${\sf Fin}^o \to L$, where $\sf Fin$ is the category of finite sets and functions. These functors form a category, the category $\sf Law$ of Lawvere theories, of which a brilliant survey is this paper by Hyland and Power https://www.sciencedirect.com/science/article/pii/S1571066107000874

Then there is an equivalence of categories between

1. The category $\sf Law$ defined as above.
2. The category of promonads on the object $\sf Fin$, regarded as an object of the bicategory $\sf Prof$ of profunctors
3. The category of finitary monads on $\sf Set$ (all sets, and functions)
4. The category of "convolution monoidal cocontinuous" monads on $[{\sf Fin,Set}]$, i.e. monads $T : [{\sf Fin,Set}] \to [{\sf Fin,Set}]$ that are colimit preserving and monoidal functors with respect to Day convolution
5. The category of "clones" (also called, "cartesian operads"), i.e. monoids with respect to a certain monoidal structure on $[{\sf Fin,Set}]$ called "substitution product".
6. The category of relative monads in the (skew)monoidal category $[{\sf Fin,Set}]$.

Here we're in the same situation of the famous parable of the elephant and the blind men: various approaches have tried to generalise, to various extents, each of these different perspectives on the notion of Lawvere theory. Some of them work better when changing the base of enrichment; some others work better when you try to internalise the notion of algebraic theory (surprisingly, the first notion of internal algebraic theory was given taking 2 as primary definition, in a brilliant paper by Johnstone and Wraith https://link.springer.com/chapter/10.1007%2FBFb0061363 ), some other work best when your abstract setting for CT is double categorical, some others are exquisitely combinatorial.

Another possible generalisation is the following (of which I know no single, comprehensive reference): $\sf Fin$ is the free category with finite coproducts over the point, and dually ${\sf Fin}^o$ is the free category with products over the point. so, call ${\sf Fin} = P1$ and replace $\sf Fin$, above, with another category $D1$, the free category with $D$-structure over the point, where $D$ s just another 2-monad on $\sf Cat$ (for example, completion under other shapes of colimits). Which parts of the correspondence above break down?

Do all these different approaches converge in a truly comprehensive, unified framework? I don't know! I would say some people do, but...