The switch is called a splitter.

Thanks for the reply, but the splitters I'm familiar with (coax for example) simply split one source to multiple outputs. There's no "routing" going on, if that's the proper term. The way I envision it, today cable box #1 is feeding TV #1. Tomorrow, cable box #2 might need to feed TV #1 because cable box #1 is already feeding TV #2.

I don't know your service area, but Spectrum (formerly Time Warner) allows video streaming with your cable package. Roku or similar devices may allow you to do what you want without needing set top boxes (cable boxes). From the 'in'/'out' you want, the word is distribution amplifier. Spectrum has gone to requiring digital adapters for basic cable if not using a set top box. Roku or similar may have connection limits defined by your cable company; high speed internet would also be a requirement.

I do have the Triple Play with Optimum in Nassau County, NY. Distribution amplifier is a term I came across while waiting for replies here, so I plan on looking into that further.

I also had Optimum in Suffolk Co. I discovered that if you connect the cable directly to the input of a smart TV, many of the channels can be picked up, although sometimes it is difficult to figure out what they are (no "guide"). The regular OTA (Over the Air) channels are easy (WCBS, WABC, etc.) so you might try that for some of your TVs.

Do you have wireless feeds today from your main box to the remote boxes like I have with my DirecTV Genie? It would honestly be easier to move the remote boxes to the TV you want to watch than to run that much HDMI cable in your house. Not to mention what that would cost. Just a thought.

I do not have main/remote boxes, just a set-top box on 5 of my TVs. I thought about running HDMI over cat5, but haven't yet fully explored the feasibility. All of this is my attempt at trimming the cable bill by eliminating 2 boxes (at $12 per month each). The ROI would have to be fairly quick (1-2 years) in order for me to pull the trigger on the project.

Looking into HDMI matrix devices, once you go past 4x4, it gets extremely expensive, as bmelz mentioned. Even if I can get a device that will suit my needs, I then need to figure out the control$!

If you have high speed internet & WiFi in your house and your cable service allows streaming of your service; cut the cord where you don't DVR. Get something similar to a ROKU device. There's more than one type of stick, offering different capabilities. What they do offer is no need for a box, no coax cable to the TV. What they don't have is capability to stream something from your DVR to the multiple TVs.

Mike C: do any of these streaming services do broadcast TV in real time?

There seems to be about a 15 second difference between what my DVR box and ROKU device shows, but yes, its in real time. Same programs and channel availability. With the Spectrum (Time Warner) streaming I can also select On Demand which I couldn't do with the Digital Adapters I had to have on 2 other TVs (3 in the house). Also when streaming, PBS & BBC America are in hi-def.; on broadcast, standard-def. Another advantage with a stick (I only know ROKU) and a cable package you can get PBS.org, FXX.com, NBC.com, CBS.com, etc. Using a stick will save $8 a month on cable bill as no more digital adapters. The downside, is if your service provider doesn't provide streaming you will only get pay services such as Netflix, Hulu, Amazon, etc.

Right now I'm leaning towards TiVo. If I get a Bolt and minis for the other TVs, my ROI should be less than 2 years. I'll have to figure out some cabling, which isn't a big issue for me. Buying directly from TiVo allows me 30 days to return no questions asked.

You might also want to check out Sling TV for network channels and some premium channels like ESPN. The basic Orange package is $20 a month. The Sling TV app is available on Roku and some smart TVs so you could use those for content at each TV. Take a look at Air TV as well. http://variety.com/2016/digital/news/sling-tv-airtv-player-netflix-1201946178/. The Air TV will combine OTA TV channels along with sling TV channel into one interface.

Update: I went with Tivo. 1 of their Bolt DVR boxes and 4 minis serve the 5 TVs we use. The Bolt I have only has 4 tuners, but that's enough to cover our needs. I used the existing coax cabling with some slight modifications to create the MoCA network needed. After returning my 5 cable boxes, I figure the decrease in my cable bill will pay off the Tivo equipment in about a year. So far the transition has been relatively easy. Tivo is a little "wonky" when compared to using cable boxes, but worth the effort nonetheless.

Thanks to all who helped me shape my decision.