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>”Why does it seem like the rate I’m burning calories on the elliptical increases during my workout?”

Does the device giving calorie readout record your heart rate? Your heart rate will increase over the first few minutes of an activity and then gradually level off even if it is at a constant rate which would give a rising value for calories burned at the same pace if it calculates calories based on HR.

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And after the workout - calories will be burned at a heightened level from baseline for several hours?

The answer to your question really depends on the type of exercise you are doing. If you are doing steady state cardio (IE your heart rate remains constant), your calories burnt will also remain constant. You may increase energy expenditure slightly as you fatigue as your movement efficiency will decrease.

If your heart rate is NOT constant, your energy expenditure will change similarly. Between your resting heart rate and 90% of your HR max, calorie usage and HR increase linearly. As your HR exceeds 90% of the max, energy expenditure increases much faster. Though most experts recommend against training at 90+% of your HR max as you risk injury and often requires extended recovery time limiting your fitness improvement as well as overall calories burned (IE going for 1 run at 95% instead of 3 runs at 70% in a week burns fewer calories and takes the same recovery time, approximately).

HR and calories burned: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4005766/
Running efficiency and fatigue: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6722636/
Metabolism during different exercises: https://www.nature.com/articles/s42255-020-0251-4

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Just to add, cardiac drift will also increase the heart rate later in a long-ish session.