There are several things that could be going on here to cause what you are describing:

• Not all daylight is equal. Time of day and time of year both affect the angle of the sun in the sky which affects how much of the sun's energy is falling on a specific amount of area, cloud cover affects how much of the sunlight reaches the ground, and even how much moisture and particulates in the air can affect overall brightness. Then there is the entire issue on a bright sunny day of which parts of the field are in direct sunlight and which parts might be in shade. The exposure values needed for each are several stops apart. Even in Manual exposure mode, metering is important! So is the Exposure Compensation setting for cameras that allow EC in M exposure mode with Auto ISO.
• The mechanical aperture lever on Nikon cameras is prone to being damaged if a lens is improperly positioned while being mounted. If the lever gets bent it will affect the actual versus selected aperture value for each shot.
• In-camera settings such as Auto ISO (and for those shooting Canon and some other camera brands - Safety Shift) can alter manually entered aperture and shutter values when set to enabled if the camera detects the scene is out of range for the selected settings.
• Different metering modes interpret the same scene differently, especially scenes in which some areas are much lighter or darker than others. That's why it is important to know the metering mode to which you have the camera set and how that affects the scene you see in your viewfinder.
• There are multiple versions of the Sigma 18-250mm lens, but all of them have a maximum aperture of f/6.3 on the long end. This means neither of your cameras can use f/5.6 when the lens is zoomed past a certain focal length. Depending on camera settings, the camera may still use the selected ISO and shutter time or it may compensate for the dimmer aperture. Check to see if the similar settings are the same for both the D610 and the D5200.
• The Sigma 18-250mm are also DX lenses, which means your D610 is shifting to crop mode and only using the center 45% of the imaging sensor. This may also be affecting metering decisions if Auto ISO (with or without EC) is enabled.

If you check the EXIF info of your previous shots with the D5200 and your more recent shots with the D610 I expect that you will find the D5200 was bumping up the ISO to compensate for the narrower maximum aperture and the D610 was not, forcing you to increase the shutter time.