Its not so much that Britain, and by extension its empire, fell behind in its industrial & economic capacity compared to other nations. Rather the reverse is true. In the 19th Century other nations, particularly the United States, Germany & France began catching up technologically, industrially and economically with the UK - diminishing the aforementioned advantage the British had in being the first nation to have mass industrialized. The same parallel can be seen as repeating itself, specifically among
emerging economies in Asia (China & India in particular) compared with the so-called "declining" economies in the developed world.
As a rule, economies that start from a lower-base have an easier time attaining economic development at a more rapid rate, as they are able to utilise the innovations and discoveries already made by developed countries and applying those methods within their own economic contexts and further; by studying the history of developed economies grew to where they are -replicating the successes found on their roads of development, avoiding the mistakes, flaws & pitfalls on their path to economic development whilst also creating their own innovations and economic ideas to further speed up economic growth within their nations at the same time. The efficacy of these processes (i.e. how fast and how effectively the grow) varies depending on the nature of the country in question. Things such as an abundance of natural resources and geographic location help greatly though more importantly - the strength and efficacy of said country's institutions, both political and economic, play a large part in determining how efficiently those countries are able use existing resources and focussing them towards attaining economic growth.
A good analogy for this can by found in
muscle hypertrophy. When a beginner starts out in working a muscle group they will after a two weeks have achieved some rapid gains in muscular development. As the muscle group continues to become more
developed however, the returns from working out those muscles will begin to diminish; requiring more effort in resistance training (e.g greater weight loads, higher sets/repetitions) to continue to develop and maintain those gains. Hence one can make a parallel between this aspect of fitness hypertrophy with the economic theory of
diminishing returns.
Quite a simplification I know, but it should give an allegorical idea of why Britain's economic
clout diminished compared to other nations (esp. America & Germany) by the 1890's.