Will all the dominoes fall?
Charcoal, 86 69 bytes
ＷＳ⊞υι≔⪫υ¶θ≔v&gt;^&lt;ηＰθ…θ⌈Ｅη⌕θιＷ№ηＫＫ«↶⊗⁺³⌕ηＫＫ≔ⅈι ≔§⪪\-//|\³⁻ⅈιη»≔⌈ＫＡθ⎚→‹θ!

Try it online! Link is to verbose version of code. Outputs a Charcoal boolean, i.e. - if all the dominoes fall, nothing if some remain. Explanation:
ＷＳ⊞υι≔⪫υ¶θ

Input the table as an array of strings and join them together.
≔v&gt;^&lt;η

Make a string of rotations for the starting domino characters. The string starts with the character for a clockwise rotation, then the one for no rotation, then an anticlockwise rotation, then a u-turn.
Ｐθ…θ⌈Ｅη⌕θι

Print the table and then up until the (last) occurrence of one the starting domino characters.
Ｗ№ηＫＫ«

While the current character is present in the table of rotations...
↶⊗⁺³⌕ηＫＫ

... pivot according to the position of the character in the table of rotations (annoyingly I have to spend an extra byte adding 3 instead of subtracting 1 because Charcoal can't pivot by negative amounts)...
≔ⅈι ≔§⪪\-//|\³⁻ⅈιη

... and, temporarily saving the current X-coordinate, erase the current character, moving in the current direction, then choose the list of valid characters (again listed in the order rotate clockwise, don't rotate, rotate anticlockwise) depending on whether the current direction is vertical or horizontal (detected via the change in X-coordinate, since Charcoal has no builtin to read the current direction).
»≔⌈ＫＡθ⎚→‹θ!

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