に従(したが)ってAs, Following, As then, According to, In accordance with

Both express change in tandem and can often be used interchangeably. にしたがって frames it as happening 'in accordance with' another, while につれて emphasizes gradual, proportional change. For example, '話すにしたがって、だんだん打ち解けてきた' and '話すにつれて、だんだん打ち解けてきた' have nearly the same meaning.

When used to mean in line with or in accordance with, にしたがって and に沿って are similar and often interchangeable, as in 規則（にしたがって／に沿って）行動する (act according to the regulations). However, に沿って can also mean along(side), as in 川に沿って歩く (walk alongside the river), while にしたがって cannot.

に伴って・に伴い describes changes in one thing due to another, in a proportional or cause-and-effect manner, as in 人口減少に伴って (along with the declining population). にしたがって can overlap in this usage, but it also extends to a broader sense of following rules or standards, as in 指示に従って (according to instructions).

をもとに and にしたがって can both be translated as according to or based on, but differ in nuance. をもとに suggests a foundation or inspiration, as in 実話をもとに書いた小説 (a novel written based on actual events). にしたがって focuses more on being in line with or following something, as in 指示にしたがって (according to instructions). It can also mark proportional changes, as in 普及にしたがって (as it spread).

に応じて and にしたがって can seem similar, but differ in nuance. に応じて means in response to something, as in アンケートの結果に応じて方針を決める (decide policy in response to the survey results). にしたがって refers to directly following guidelines or standards, as in 規則に従って (in accordance with the regulations).

Both に則って and にしたがって mean according to or in accordance with. に則って is more formal and often used in technical or legal contexts, as in 法律に則って (in accordance with the law). にしたがって has broader usage: besides according to, it can also express sequential or proportional relationships, as in 普及にしたがって (as it spread).

れる・られる + ままに implies doing something in accordance with external directions, often with little control, as in 指示されるがままに (exactly as instructed). にしたがって also means in accordance with, as in 規則に従って (in accordance with the regulations), but does not necessarily imply lack of volition.

Both are formal expressions meaning 'in accordance with (A)', but differ in usage. にもとづいて shows the foundation for an action, e.g. 法律にもとづいて判断する 'make a decision based on the law.' にしたがって can overlap, but often shows related change, e.g. 年を取るにしたがって目が悪くなっていく 'Eyesight deteriorates with age.' In this case, もとづいて would not be used.

Both relate to the idea of being in accordance with something else, but with different nuances and usage. に合わせて・に合った follows a noun and implies adjustment to suit a person or situation, e.g. 相手に合わせて話し方を変える ('Adjust your way of speaking to the other person'), while にしたがって shows something happening in response or proportion to another change, e.g. 年を取るにしたがって体力が落ちる ('As you age, your stamina declines').

Both can translate as 'according to', but differ in usage. にしたがって means 'in accordance with' or 'following' something, e.g. 説明書に従って組み立てた ('I assembled it according to the manual.') によると・によれば expresses reported information, e.g. 説明書によると操作は簡単です ('According to the manual, the operation is simple.')