Huge analog switching matrix
So, when you say

100 pins connectors and has to be capable of arbitrarily connect each pin on the first connector to each pin of the second.

and imagine one connector to be an input vector \$U\in \mathbb F^{100}\$, the other to be a result vector \$V \in \mathbb F^{100}\$, then your whole device would need to be a 100×100 permutation matrix \$\Pi\in \mathbb F^{100\times 100}\$. (i.e. a matrix where each row and column has a single unity element, the rest being zero):
$$V=\Pi U\tag1\label{full}$$
You can straightforward implement that using any kind of switches.
And such a matrix has 10,000 entries that can be 0 or 1 – i.e. 10⁴ switches; relays that you control from a PC would probably be wisest, simply to avoid human error.
Now, you say:

The signal going through the board, once ALL the connections are set, will be on only a subset (4) lines at any given time

Well, that makes it easier:
Instead of having \$\eqref{full}\$, you can imagine an intermediate result vector \$W\in \mathbb F^{4}\$.
If you don't even &quot;blow this back up&quot; to the full 100 outputs in \$V\$, and I don't see a reason for that, your measurement devices doesn't seem to care whether it scans 100 channels or just the 4, that boils down to
$$\tilde V = \mathbf PU,$$
with \$\mathbf P\in  \mathbb F^{100\times 4}\$, i.e. only 400 switches instead of 10,000.
If you can further restrict that (for example, there's only specific index subsets from which the first of your four outputs can come), then you might further simplify the problem: For example, four 25×1 matrices are still way easier than one 100×4 one!
An order of 400 switches still isn't cheap, and now you'll have to switch dynamically when you want to measure a different 4-set of channels. However, signal relays (exactly for switched-circuit telephony historical reasons) are comparatively cheap (A list).
Considering coil voltages and currents of 5 V, 40 mA typ., you might just directly drive these off open-drain or open-collector shift registers for a couple of cents, e.g. STPIC6D595, which you can in turn directly attach to a 5V bitbang/SPI-driver (which exist either as microcontrollers or as USB-to-SPI converters, for example).
Rough calculation of the board you'll need: The relay I linked to has an area of 11×21 mm², and you need a 10×6 mm² shift register for every 8 relays.
That makes a total area of 400·231 mm² + 50·60 mm² = 95400 mm². If you're a bit symmetrical about arranging the 8 relays head-to-head and the shift register at the end of each such row, you end up with a ca 17 cm wide board; if you put 32 rows on a board, that will be, including auxillary things (connectors etc) be 40 cm tall. Manufactured in China, five boards of that size will set you back by ca 40 to 80€ incl. shipping.
Adding 440 relays and 55 shift registers (10% overorder in case you damage anything) for about 300 € (not including taxes); throw in 50€ for connectors, power supply, LEDs and stuff, this is a 400€ project.
Compare that to the 10,000 switches: rather cheap and easy!
(in case you're a PhD candidate physicist or similar: get a student research assistant to design, and assemble, and test, and write a minimal control software for that board. Pay her or him well, i.e. overprovision her hours if necessary. These kinds of devices, if done well, usually have long-term utility after the end of your PhD. Add a sticker to it that points to some public documentation, and put all design files under CERN's open hardware license v2, and put everything on the internet, e.g. github, so that the next physicist doesn't reinvent the wheel. People have been cited for less in other people's papers!)